Geometry and Trigonometry Alıştırma Soruları — SAT — Sayfa 6

Soru 101Soru →

In right triangle $XYZ$ , the measure of angle $Y$ is $90^\circ$ . If $\cos(X) + \sin(Z) = 1.6$ , what is the value of $\sin(X)$ ?

Cevabı ve açıklamayı göster

Cevap: 0.6

Cevap

0.6

The two acute angles in a right triangle sum to

90^\circ

, which means they are complementary. Therefore, the co-function identity

\sin(Z) = \cos(X)

applies. Substituting this relationship into the equation

\cos(X) + \sin(Z) = 1.6

gives

2\cos(X) = 1.6

, leading to

\cos(X) = 0.8

. Finally, using the Pythagorean identity

\sin^2(X) + \cos^2(X) = 1

, we find

\sin(X) = \sqrt{1 - 0.8^2} = 0.6

.

Adım Adım Çözüm

1

Determine the relationship between angles

X

and

Z

.

X + Z = 90^\circ

Since the sum of the angles in right triangle

XYZ

is

180^\circ

and angle

Y

is

90^\circ

, the two acute angles

X

and

Z

must sum to

90^\circ

, making them complementary.

2

Use the co-function identity to relate

\sin(Z)

and

\cos(X)

.

\sin(Z) = \cos(X)

For complementary angles, the sine of one angle is equal to the cosine of the other.

3

Substitute the identity into the given equation to solve for

\cos(X)

.

\cos(X) = 0.8

Replacing

\sin(Z)

with

\cos(X)

in

\cos(X) + \sin(Z) = 1.6

yields

2\cos(X) = 1.6

, so

\cos(X) = 0.8

.

4

Solve for

\sin(X)

using the Pythagorean identity.

\sin(X) = 0.6

Since

\sin^2(X) + \cos^2(X) = 1

, we have

\sin^2(X) + (0.8)^2 = 1

, which gives

\sin^2(X) = 1 - 0.64 = 0.36

. Since

X

is an acute angle, its sine must be positive, so

\sin(X) = 0.6

.

Anahtar Kavram

Co-function and Pythagorean trigonometric identities in right triangles

Soru 102Soru →

In the $xy$ -plane, a particle moves along a circular path centered at the origin. The particle starts at the point $(3, 0)$ and travels counterclockwise along the circle. If the particle stops after traveling an arc length of $5\pi$ units, what is the measure of the central angle subtended by the particle's path, in degrees?

150^\circ

225^\circ

300^\circ

600^\circ

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Cevap: $300^\circ$

Cevap

300^\circ

The radius of the circular path is

3

units since it starts at

(3,0)

and is centered at the origin. Using the arc length formula

s = r\theta

where

s = 5\pi

is the arc length and

r = 3

is the radius, the central angle is

\theta = \frac{5\pi}{3}

radians. To convert this angle to degrees, we multiply by

\frac{180^\circ}{\pi}

, which yields

\frac{5\pi}{3} \times \frac{180^\circ}{\pi} = 300^\circ

.

Adım Adım Çözüm

1

Determine the radius of the circular path.

r = 3

Since the path is centered at the origin

(0, 0)

and starts at

(3, 0)

, the distance from the center to the path (the radius) is

3

units.

2

Calculate the central angle in radians using the arc length formula.

\theta = \frac{5\pi}{3}

radians

The relationship between arc length

s

, radius

r

, and central angle

\theta

in radians is given by

s = r\theta

. Substituting

s = 5\pi

and

r = 3

gives

5\pi = 3\theta

, which simplifies to

\theta = \frac{5\pi}{3}

.

3

Convert the central angle from radians to degrees.

300^\circ

To convert radians to degrees, multiply the angle in radians by

\frac{180^\circ}{\pi}

. This gives

\theta = \frac{5\pi}{3} \times \frac{180^\circ}{\pi} = 5 \times 60^\circ = 300^\circ

.

Anahtar Kavram

Using the arc length formula

s = r\theta

to find an angle in radians and converting that angle to degrees.

Soru 103Soru →

In the $xy$ -plane, a circle with its center at the origin contains the point $P(6, -2\sqrt{3})$ . Point $P$ is rotated counterclockwise about the origin by an angle of $\frac{5\pi}{6}$ radians to a new position $Q$ . What are the coordinates of point $Q$ ?

(0, -4\sqrt{3})

(-4\sqrt{3}, 0)

(-2\sqrt{3}, 6)

(-6, 2\sqrt{3})

Cevabı ve açıklamayı göster

Cevap: $(-2\sqrt{3}, 6)$

Cevap

The coordinates of point

Q

are

(-2\sqrt{3}, 6)

.

The correct coordinates are found by calculating the circle's radius

R = 4\sqrt{3}

and identifying that the initial point

P(6, -2\sqrt{3})

has an angle of

-\frac{\pi}{6}

radians. Rotating counterclockwise by

\frac{5\pi}{6}

radians results in a new angle of

\frac{2\pi}{3}

radians (

120^\circ

). Using circular trigonometry,

x = R\cos(120^\circ) = -2\sqrt{3}

and

y = R\sin(120^\circ) = 6

, giving the point

(-2\sqrt{3}, 6)

.

Adım Adım Çözüm

1

Find the radius

R

of the circle using the coordinates of point

P(6, -2\sqrt{3})

.

R = \sqrt{6^2 + (-2\sqrt{3})^2} = \sqrt{36 + 12} = \sqrt{48} = 4\sqrt{3}

.

The radius of the circle is needed to determine the coordinates of the rotated point

Q

on the same circle.

2

Determine the initial angle

\theta_P

of point

P

in standard position.

Since

\cos(\theta_P) = \frac{6}{4\sqrt{3}} = \frac{\sqrt{3}}{2}

and

\sin(\theta_P) = \frac{-2\sqrt{3}}{4\sqrt{3}} = -\frac{1}{2}

, the point lies in the fourth quadrant with

\theta_P = -\frac{\pi}{6}

radians (or

330^\circ

).

Finding the initial position in standard angle form allows us to apply the rotation.

3

Calculate the new angle

\theta_Q

after the counterclockwise rotation of

\frac{5\pi}{6}

radians.

\theta_Q = -\frac{\pi}{6} + \frac{5\pi}{6} = \frac{4\pi}{6} = \frac{2\pi}{3}

radians.

Adding the counterclockwise rotation angle to the initial angle gives the new standard position angle of the terminal ray.

4

Convert the new angle to degrees and compute the coordinates of point

Q

.

Converting the angle gives

\frac{2\pi}{3} \times \frac{180^\circ}{\pi} = 120^\circ

. The coordinates of

Q

are

x_Q = 4\sqrt{3} \cos(120^\circ) = -2\sqrt{3}

and

y_Q = 4\sqrt{3} \sin(120^\circ) = 6

.

Converting to degrees allows the use of standard trigonometric values to find the exact coordinates.

Anahtar Kavram

Radian-to-degree conversion and application of rotations on a coordinate plane using circular trigonometry.

Alternatif Yöntem

Instead of working entirely in radians, convert the coordinates of

P

to degrees first. Since

\tan(\theta_P) = -\frac{\sqrt{3}}{3}

in Quadrant IV,

\theta_P = -30^\circ

. The rotation of

\frac{5\pi}{6}

radians is converted to degrees:

\frac{5\pi}{6} \times \frac{180^\circ}{\pi} = 150^\circ

. Adding these gives the new angle

\theta_Q = -30^\circ + 150^\circ = 120^\circ

. Finally, compute the coordinates using

x = R\cos(120^\circ)

and

y = R\sin(120^\circ)

.

Tahmini Süre:3m 0s

Soru 104Soru →

A metallic plate is in the shape of a trapezoid. The parallel sides of the plate have lengths of $9$ centimeters and $30$ centimeters. The two non-parallel sides have lengths of $10$ centimeters and $17$ centimeters. What is the area, in square centimeters, of the metallic plate?

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Cevap: 156

Cevap

156

To find the area of the trapezoid, we can find its height by drawing perpendicular lines from the vertices of the shorter base to the longer base. This splits the trapezoid into a rectangle of width

9\text{ cm}

and two right triangles with hypotenuses of

10\text{ cm}

and

17\text{ cm}

. The sum of the bases of these two triangles is

30 - 9 = 21\text{ cm}

. Letting their bases be

x

and

y

(where

x + y = 21

), we apply the Pythagorean theorem:

h^2 + x^2 = 100

and

h^2 + y^2 = 289

. Subtracting the equations yields

y^2 - x^2 = 189

. Since

y^2 - x^2 = (y - x)(y + x)

, we have

21(y - x) = 189

, which simplifies to

y - x = 9

. Solving the system

x + y = 21

and

y - x = 9

yields

x = 6

and

y = 15

. Using the Pythagorean triple

6

-

8

-

10

, the height

h

is

8\text{ cm}

. The area is

\frac{1}{2}(9 + 30)(8) = 156

square centimeters.

Adım Adım Çözüm

1

Drop perpendicular lines (heights) from the endpoints of the shorter base to the longer base.

The trapezoid is decomposed into a central rectangle with a width of

9

centimeters and two right triangles with bases

x

and

y

and hypotenuses of

10

centimeters and

17

centimeters, respectively.

This sets up a system of equations relating the heights and bases of the right triangles.

2

Determine the relationship between the bases of the two right triangles.

The sum of the bases of the two right triangles is

x + y = 30 - 9 = 21

centimeters.

The sum of the bases of the right triangles plus the width of the rectangle equals the total length of the longer base of the trapezoid.

3

Set up equations using the Pythagorean theorem for the two right triangles.

h^2 + x^2 = 100

and

h^2 + y^2 = 289

.

Both right triangles share the same height

h

of the trapezoid.

4

Solve for the difference between the two triangle bases.

Subtracting the first equation from the second gives

y^2 - x^2 = 189

. Factoring yields

(y - x)(y + x) = 189

. Substituting

y + x = 21

yields

21(y - x) = 189 \implies y - x = 9

.

Subtracting the equations eliminates the height variable, allowing us to find the difference between the bases.

5

Solve the system of linear equations for

x

and

y

.

Adding

x + y = 21

and

y - x = 9

gives

2y = 30 \implies y = 15

. Substituting back gives

x = 6

.

This determines the exact base segments of both right triangles.

6

Calculate the height of the trapezoid.

h = \sqrt{10^2 - 6^2} = \sqrt{64} = 8

centimeters.

The height of the trapezoid is required to calculate its area.

7

Compute the area of the trapezoid.

\text{Area} = \frac{1}{2}(9 + 30)(8) = 156

square centimeters.

This uses the standard formula for the area of a trapezoid.

Anahtar Kavram

Decomposing a non-isosceles trapezoid into a rectangle and two right triangles to solve for the height using systems of quadratic equations derived from the Pythagorean theorem.

Tahmini Süre:1m 30s

Soru 105Soru →

In right triangle $ABC$ , the measure of angle $C$ is $90^\circ$ . Point $D$ lies on side $AC$ and point $E$ lies on hypotenuse $AB$ such that segment $DE$ is perpendicular to $AB$ . The length of segment $AE$ is $x$ , the length of segment $AD$ is $x + 1$ , the length of segment $CD$ is $2$ , and the length of segment $BE$ is $5$ . What is the value of $x$ ?

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Cevap: 3

Cevap

3

By identifying that triangle

AED

is similar to triangle

ACB

(due to shared angle

A

and right angles at

E

and

C

), we can set up the proportion

\frac{AE}{AC} = \frac{AD}{AB}

. Expressing the segments as

AC = x + 3

and

AB = x + 5

leads to the equation

\frac{x}{x + 3} = \frac{x + 1}{x + 5}

. Solving this equation yields

x = 3

.

Adım Adım Çözüm

1

Establish similarity between triangles

AED

and

ACB

.

\triangle AED \sim \triangle ACB

Both triangles share the acute angle

A

(

\angle EAD = \angle CAB

) and both have a right angle (

\angle AED = \angle ACB = 90^\circ

), satisfying the Angle-Angle (AA) similarity criterion.

2

Write the proportion of corresponding sides.

\frac{AE}{AC} = \frac{AD}{AB}

In similar triangles, the ratios of the lengths of corresponding sides are equal.

3

Express the total side lengths of triangle

ABC

using segment addition.

AC = x + 3

and

AB = x + 5

Since

D

is on

AC

,

AC = AD + CD = (x + 1) + 2 = x + 3

. Since

E

is on

AB

,

AB = AE + BE = x + 5

.

4

Substitute the algebraic expressions into the side ratio proportion.

\frac{x}{x + 3} = \frac{x + 1}{x + 5}

Substituting

AE = x

,

AD = x + 1

,

AC = x + 3

, and

AB = x + 5

into the similarity proportion.

5

Solve the proportion for

x

.

x = 3

Cross-multiplying gives

x(x + 5) = (x + 1)(x + 3) \implies x^2 + 5x = x^2 + 4x + 3

. Subtracting

x^2

and

4x

from both sides results in

x = 3

.

Anahtar Kavram

Identifying similar right triangles via the AA similarity criterion and solving resulting algebraic proportions.

Soru 106Soru →

An acute angle $\theta$ in a right triangle satisfies the equation $\cos(\theta) = \frac{5}{13}$ . What is the value of $\tan(90^\circ - \theta)$ ?

\frac{12}{5}

\frac{12}{13}

\frac{5}{12}

\frac{5}{13}

Cevabı ve açıklamayı göster

Cevap: $\frac{5}{12}$

Cevap

\frac{5}{12}

The value

\frac{5}{12}

is correct because in any right triangle with acute angle

\theta

, the complementary angle is

90^\circ - \theta

. The side adjacent to

\theta

becomes the side opposite to

90^\circ - \theta

, and the side opposite to

\theta

becomes the side adjacent to

90^\circ - \theta

. Given

\cos(\theta) = \frac{5}{13}

, the adjacent side is

5

and the hypotenuse is

13

. The remaining side is

12

by the Pythagorean theorem. Therefore,

\tan(90^\circ - \theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{12}

.

Adım Adım Çözüm

1

Identify the relationship between the acute angles in a right triangle.

The two acute angles of a right triangle sum to

90^\circ

. Therefore, the other acute angle is

90^\circ - \theta

.

Since the sum of angles in any triangle is

180^\circ

and one angle is

90^\circ

, the remaining two angles must sum to

90^\circ

.

2

Determine the side lengths of the right triangle based on the given ratio.

Since

\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{5}{13}

, we can let the adjacent side to

\theta

be

5

and the hypotenuse be

13

. Using the Pythagorean theorem, the opposite side to

\theta

is

\sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12

.

The Pythagorean theorem relates the sides of a right triangle:

\text{opposite}^2 + \text{adjacent}^2 = \text{hypotenuse}^2

.

3

Calculate the value of

\tan(90^\circ - \theta)

using the side lengths.

For the angle

(90^\circ - \theta)

, the opposite side is the side adjacent to

\theta

(length

5

), and the adjacent side is the side opposite to

\theta

(length

12

). Thus,

\tan(90^\circ - \theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{12}

.

By definition, the tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.

Anahtar Kavram

Complementary angle trigonometric relationships and Pythagorean triple side determinations in right triangles.

Alternatif Yöntem

Alternatively, we can use trigonometric identities:

\tan(90^\circ - \theta) = \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}

. Since

\sin(\theta) = \sqrt{1 - \cos^2(\theta)} = \sqrt{1 - \left(\frac{5}{13}\right)^2} = \frac{12}{13}

, we get

\cot(\theta) = \frac{5/13}{12/13} = \frac{5}{12}

.

Tahmini Süre:1m 15s

Soru 107Soru →

Two connected gears, Gear A and Gear B, rotate together such that the belt connecting them does not slip. The radius of Gear A is $15$ centimeters and the radius of Gear B is $9$ centimeters. If Gear A rotates through a central angle of $\frac{4\pi}{15}$ radians, Gear B rotates through a central angle of $x$ degrees. What is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 80

Cevap

80

The arc length

s

that a point on the belt travels is given by the product of the radius and the angle in radians of Gear A:

s = 15 \times \frac{4\pi}{15} = 4\pi

cm. Since the belt does not slip, Gear B rotates through the same arc length. The angle of Gear B in radians is

\theta = \frac{4\pi}{9}

radians. To convert this angle to degrees, multiply by

\frac{180}{\pi}

to get

\frac{4\pi}{9} \times \frac{180}{\pi} = 80

degrees.

Adım Adım Çözüm

1

Calculate the arc length of the rotation for Gear A using the formula

s = r\theta

.

s = 4\pi

centimeters

To find the distance a point on the belt travels, which is shared by both gears.

2

Determine the rotation angle of Gear B in radians using the arc length and Gear B's radius.

\theta_B = \frac{4\pi}{9}

radians

Because the belt does not slip, Gear B must rotate by the same linear arc length as Gear A.

3

Convert the angle of Gear B from radians to degrees by multiplying by

\frac{180}{\pi}

.

x = 80

To find the measure of the angle in degrees as requested by the question.

Anahtar Kavram

Converting central angles between radians and degrees in the context of arc lengths of connected circles.

Soru 108Soru →

An artist designs a logo consisting of a rectangle and an isosceles triangle. The rectangle has a length of $12$ centimeters and a width of $8$ centimeters. The base of the isosceles triangle is one of the $8$ -centimeter sides of the rectangle, and the vertex of the triangle lies outside the rectangle. If the total area of the logo is $120$ square centimeters, what is the height, in centimeters, of the triangle?

3

4

6

12

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

The total area of the logo is the sum of the area of the rectangle and the area of the triangle. The area of the rectangle is the product of its length and width:

12 \times 8 = 96

square centimeters. Subtracting the area of the rectangle from the total area gives the area of the triangle:

120 - 96 = 24

square centimeters. The area of a triangle is given by the formula

\frac{1}{2} \times \text{base} \times \text{height}

. The base of the triangle is one of the

8

-centimeter sides of the rectangle. Substituting the base and area into the formula gives

24 = \frac{1}{2} \times 8 \times \text{height}

, which simplifies to

24 = 4 \times \text{height}

. Dividing both sides by

4

yields a height of

6

centimeters.

Adım Adım Çözüm

1

Calculate the area of the rectangle.

The area of the rectangle is

12 \times 8 = 96

square centimeters.

The total area of the logo is composite, so we must find the area of the rectangular portion first.

2

Calculate the area of the triangle.

The area of the triangle is

120 - 96 = 24

square centimeters.

Subtracting the rectangle's area from the total area of the logo yields the remaining area occupied by the triangle.

3

Solve for the height of the triangle using the area formula.

The height of the triangle is

6

centimeters.

The base of the triangle is

8

centimeters. Using the formula

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

, we set up the equation

24 = \frac{1}{2} \times 8 \times \text{height}

, which simplifies to

24 = 4 \times \text{height}

, giving a height of

6

centimeters.

Anahtar Kavram

The area of a composite shape is the sum of the areas of its simpler component shapes.

Soru 109Soru →

In a circle with center $O$ , chord $AB$ has length $12\sqrt{3}$ . A radius $OC$ is perpendicular to chord $AB$ and intersects $AB$ at point $D$ . If $CD = 6$ , what is the area of the sector of the circle bounded by radii $OA$ , $OB$ , and the minor arc $AB$ ?

18\pi

24\pi

48\pi

144\pi

Cevabı ve açıklamayı göster

Cevap: $48\pi$

Cevap

The correct answer is the option representing

48\pi

.

The correct answer is

48\pi

because the radius of the circle is determined to be

12

using the Pythagorean theorem, and the central angle of the sector is

120^\circ

. The area is then

\frac{120}{360} \times \pi (12)^2 = 48\pi

.

Adım Adım Çözüm

1

Determine the length of the segment from the midpoint of the chord to its endpoints.

AD = 6\sqrt{3}

A radius perpendicular to a chord bisects the chord, so

D

is the midpoint of chord

AB

.

2

Express the distance from the center

O

to the chord intersection

D

in terms of the radius

R

and set up the Pythagorean theorem for right triangle

\triangle ODA

.

R^2 = (R - 6)^2 + (6\sqrt{3})^2

The radius

OC

has length

R

, making

OD = OC - CD = R - 6

. Since

\triangle ODA

is a right triangle, we can apply the Pythagorean theorem.

3

Solve the equation for the radius

R

.

R = 12

Expanding the equation gives

R^2 = R^2 - 12R + 36 + 108

, which simplifies to

12R = 144

.

4

Find the central angle

\angle AOB

.

\angle AOB = 120^\circ

In right triangle

\triangle ODA

, the cosine of

\angle AOD

is

\frac{OD}{OA} = \frac{6}{12} = \frac{1}{2}

, which means

\angle AOD = 60^\circ

. The total central angle is

\angle AOB = 2 \times \angle AOD = 120^\circ

.

5

Calculate the area of the sector bounded by

OA

,

OB

, and the minor arc

AB

.

48\pi

The area of the sector is the fraction of the circle's total area corresponding to the central angle:

\frac{120}{360} \times \pi (12)^2 = 48\pi

.

Anahtar Kavram

Using perpendicular bisector chord properties and right triangle trigonometry to determine circle sector area

Tahmini Süre:3m 0s

Soru 110Soru →

The measure of angle $A$ is $45^\circ$ greater than the measure of angle $B$ . If the measure of angle $B$ is $\frac{5\pi}{12}$ radians, what is the measure of angle $A$ , in degrees?

Cevabı ve açıklamayı göster

Cevap: 120

Cevap

120

The correct answer is 120. First, convert the measure of angle

B

from radians to degrees:

\frac{5\pi}{12} \times \frac{180}{\pi} = \frac{5 \times 180}{12} = 5 \times 15 = 75^\circ

. Since angle

A

is

45^\circ

greater than angle

B

, add

45^\circ

to the measure of angle

B

:

75^\circ + 45^\circ = 120^\circ

.

Adım Adım Çözüm

1

Convert the measure of angle

B

from radians to degrees.

Angle

B

has a measure of

75^\circ

.

To convert from radians to degrees, multiply the radian measure by

\frac{180}{\pi}

.

2

Calculate the measure of angle

A

by adding

45^\circ

to the measure of angle

B

.

Angle

A

has a measure of

120^\circ

.

It is given that the measure of angle

A

is

45^\circ

greater than the measure of angle

B

.

Anahtar Kavram

To convert an angle from radians to degrees, multiply the angle measure in radians by

\frac{180}{\pi}

.

Tahmini Süre:1m 0s

Soru 111Soru →

In a circle with center $O$ , $AB$ is a diameter. Point $C$ lies on the circle such that the measure of arc $AC$ is $\frac{5\pi}{9}$ radians. Point $D$ lies on the circle such that chord $AC$ is parallel to segment $OD$ , and points $C$ and $D$ lie on the same side of diameter $AB$ . What is the measure, in radians, of angle $COD$ ?

\frac{5\pi}{18}

\frac{2\pi}{9}

\frac{4\pi}{9}

\frac{13\pi}{18}

Cevabı ve açıklamayı göster

Cevap: $\frac{2\pi}{9}$

Cevap

The correct answer is

\frac{2\pi}{9}

radians.

The correct answer is

\frac{2\pi}{9}

radians. First, the central angle

\angle AOC

has a measure of

\frac{5\pi}{9}

radians because it subtends an arc of the same measure. Since

OA

and

OC

are both radii of the circle, triangle

AOC

is isosceles with

OA = OC

, which means the base angles are equal:

\angle OAC = \angle OCA = \frac{\pi - 5\pi/9}{2} = \frac{2\pi}{9}

radians. Since chord

AC

is parallel to segment

OD

and the diameter

AB

acts as a transversal line, the corresponding angles

\angle OAC

and

\angle BOD

are equal, so

\angle BOD = \frac{2\pi}{9}

radians. Finally, because

A

,

O

, and

B

form a straight line, the angles along the diameter must sum to

\pi

radians:

\angle COD = \pi - \angle AOC - \angle BOD = \pi - \frac{5\pi}{9} - \frac{2\pi}{9} = \frac{2\pi}{9}

radians.

Adım Adım Çözüm

1

Identify the measure of the central angle

\angle AOC

from the given arc measure.

The central angle

\angle AOC = \frac{5\pi}{9}

radians.

The measure of an arc in radians is equal to the measure of its subtended central angle.

2

Determine the measure of the inscribed angle

\angle OAC

using the properties of triangle

AOC

.

\angle OAC = \frac{2\pi}{9}

radians.

Since

OA

and

OC

are radii, triangle

AOC

is isosceles with

OA = OC

, meaning

\angle OAC = \angle OCA

. The sum of angles in a triangle is

\pi

radians, so

\angle OAC = \frac{\pi - \angle AOC}{2} = \frac{\pi - \frac{5\pi}{9}}{2} = \frac{2\pi}{9}

radians.

3

Use the parallel lines

AC

and

OD

to find the measure of angle

\angle BOD

.

\angle BOD = \frac{2\pi}{9}

radians.

Since chord

AC

is parallel to segment

OD

and diameter

AB

is a transversal line, the corresponding angles

\angle OAC

and

\angle BOD

are equal.

4

Calculate the measure of angle

COD

using the angles along the diameter

AB

.

\angle COD = \frac{2\pi}{9}

radians.

Points

A

,

O

, and

B

lie on a straight line, so the angles

\angle AOC

,

\angle COD

, and

\angle BOD

must sum to

\pi

radians. Therefore,

\angle COD = \pi - \angle AOC - \angle BOD = \pi - \frac{5\pi}{9} - \frac{2\pi}{9} = \frac{2\pi}{9}

radians.

Anahtar Kavram

Angle relationships in circles, including central angles, inscribed angles in isosceles triangles, and parallel line transversal properties.

Tahmini Süre:2m 30s

Soru 112Soru →

In the $xy$ -plane, triangle $ABC$ is similar to triangle $DEF$ , where vertices $A$ , $B$ , and $C$ correspond to vertices $D$ , $E$ , and $F$ , respectively. The vertices of triangle $ABC$ are $A(0, 0)$ , $B(6, 0)$ , and $C(0, 8)$ . The vertices of triangle $DEF$ are $D(2, 1)$ , $E(11, 1)$ , and $F(2, y)$ , where $y > 1$ . What is the value of $y$ ?

7

12

13

16

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

13

The correct answer is 13. Since triangle

ABC

is similar to triangle

DEF

, the ratio of their corresponding sides is constant. Side

AB

has length

6

, and the corresponding side

DE

has length

11 - 2 = 9

. This gives a scale factor of

\frac{9}{6} = 1.5

. Applying this scale factor to the vertical side

AC

(length

8

) yields a length of

12

for the corresponding side

DF

. Since

D

has coordinates

(2, 1)

and

F

has coordinates

(2, y)

with

y > 1

, the length of

DF

is

y - 1 = 12

, which solves to

y = 13

.

Adım Adım Çözüm

1

Determine the lengths of the corresponding sides of triangle

ABC

.

The vertices of triangle

ABC

are

A(0, 0)

,

B(6, 0)

, and

C(0, 8)

. The length of horizontal side

AB

is

6 - 0 = 6

, and the length of vertical side

AC

is

8 - 0 = 8

.

To find the similarity ratio between the two triangles, we need the lengths of the sides of the first triangle.

2

Determine the length of the corresponding side

DE

of triangle

DEF

.

The vertices of side

DE

are

D(2, 1)

and

E(11, 1)

. Since they share the same

y

-coordinate, this is a horizontal segment with a length of

11 - 2 = 9

.

Since vertex

D

corresponds to

A

and vertex

E

corresponds to

B

, side

DE

corresponds to side

AB

.

3

Calculate the scale factor between the similar triangles.

The scale factor from triangle

ABC

to triangle

DEF

is

\frac{DE}{AB} = \frac{9}{6} = 1.5

.

Similar triangles have corresponding side lengths that are proportional.

4

Find the length of side

DF

and the coordinate

y

.

Side

DF

corresponds to side

AC

. The length of

DF

is

1.5 \times AC = 1.5 \times 8 = 12

. Since

D

is at

(2, 1)

and

F

is at

(2, y)

, the vertical distance is

y - 1 = 12

, which gives

y = 13

.

We apply the scale factor to the corresponding side length and use the coordinate of

D

to solve for the coordinate

y

of

F

.

Anahtar Kavram

Using the properties of similar triangles and coordinates in the plane to solve for unknown lengths and coordinates.

Soru 113Soru →

A rotating beacon rotates at a constant rate of $160^\circ$ per second. Through how many radians does the beacon rotate in $4.5$ seconds?

\frac{4}{\pi}

2\pi

4\pi

\frac{32\pi}{9}

Cevabı ve açıklamayı göster

Cevap: $4\pi$

Cevap

The correct answer is the value of

4\pi

radians.

The total angle of rotation in degrees is found by multiplying the rate of rotation by the time elapsed:

160^\circ/\text{second} \times 4.5\text{ seconds} = 720^\circ

. To convert this angle from degrees to radians, multiply the degree measure by the conversion factor

\frac{\pi}{180^\circ}

. This yields

720 \times \frac{\pi}{180} = 4\pi

radians.

Adım Adım Çözüm

1

Calculate the total angle of rotation in degrees by multiplying the rate of rotation by the time.

160^\circ \times 4.5 = 720^\circ

To find the total angular displacement, we multiply the constant angular speed by the duration of the rotation.

2

Convert the total angle from degrees to radians by multiplying by the conversion factor

\frac{\pi}{180^\circ}

.

720^\circ \times \frac{\pi}{180^\circ} = 4\pi

radians

Since

180^\circ

is equivalent to

\pi

radians, multiplying by

\frac{\pi}{180^\circ}

converts the angle to radians.

Anahtar Kavram

Radian-degree conversion

Alternatif Yöntem

Convert the speed of rotation to radians per second first:

160^\circ \times \frac{\pi}{180^\circ} = \frac{8\pi}{9}

radians per second. Then multiply by the time elapsed:

\frac{8\pi}{9} \times 4.5 = 4\pi

radians.

Tahmini Süre:1m 0s

Soru 114Soru →

A surveyor maps a triangular plot of land, $LMN$ . A boundary line segment is drawn from point $P$ on side $LN$ to point $Q$ on side $LM$ , creating a smaller triangular section $LPQ$ . The measure of angle $LMN$ is equal to the measure of angle $LPQ$ . The surveyed lengths are $LM = 18$ meters, $LP = 8$ meters, and $LQ = 12$ meters. What is the length, in meters, of segment $PN$ ?

Cevabı ve açıklamayı göster

Cevap: 19

Cevap

19

By identifying that triangle

LMN

and triangle

LPQ

share the angle at vertex

L

and have another pair of congruent angles (

\angle LMN = \angle LPQ

), we establish similarity between the two triangles:

\triangle LMN \sim \triangle LPQ

. Using the proportional relationship of their corresponding sides, we write

\frac{LM}{LP} = \frac{LN}{LQ}

. Substituting the given values

LM = 18

,

LP = 8

, and

LQ = 12

yields

\frac{18}{8} = \frac{LN}{12}

, which solves to

LN = 27

. Finally, subtracting

LP = 8

from the total length of segment

LN

gives the length of segment

PN

as

19

meters.

Adım Adım Çözüm

1

Identify similar triangles.

Triangle

LMN

is similar to triangle

LPQ

(

\triangle LMN \sim \triangle LPQ

).

They share angle

L

(

\angle MLN = \angle PLQ

) and we are given that

\angle LMN = \angle LPQ

. By the Angle-Angle (AA) similarity criterion, the two triangles are similar.

2

Set up the ratio of corresponding sides.

\frac{LM}{LP} = \frac{LN}{LQ}

Corresponding sides of similar triangles are proportional.

3

Solve for the length of side

LN

.

LN = 27

meters

Substituting

LM = 18

,

LP = 8

, and

LQ = 12

into the proportion yields

\frac{18}{8} = \frac{LN}{12}

. Solving for

LN

gives

LN = 12 \times \frac{18}{8} = 27

.

4

Calculate the length of segment

PN

.

PN = 19

meters

Point

P

lies on segment

LN

, so the length of segment

PN

is the difference between

LN

and

LP

, which is

27 - 8 = 19

.

Anahtar Kavram

Triangle similarity using the Angle-Angle (AA) criterion and proportional side ratios.

Soru 115Soru →

A pendulum swings through an angle of $40^\circ$ , and the tip of the pendulum travels an arc of length $8\pi$ inches. What is the length of the pendulum, in inches?

Cevabı ve açıklamayı göster

Cevap: 36

Cevap

36

To find the length of the pendulum, which represents the radius

r

of the circular path it sweeps, we can use the arc length formula

s = r\theta

, where

s

is the arc length and

\theta

is the central angle in radians. First, convert the given angle from degrees to radians:

\theta = 40^\circ \times \frac{\pi}{180^\circ} = \frac{2\pi}{9}

radians. Next, substitute the arc length

s = 8\pi

and the angle

\theta = \frac{2\pi}{9}

into the formula:

8\pi = r \left(\frac{2\pi}{9}\right)

. Solving for

r

by multiplying both sides by

\frac{9}{2\pi}

gives

r = 36

. Alternatively, you can use the ratio of the sector's central angle to the total angle of a circle:

\frac{40^\circ}{360^\circ} = \frac{1}{9}

. This means the arc length is

\frac{1}{9}

of the circumference of the circle:

8\pi = \frac{1}{9}(2\pi r)

. Dividing both sides by

2\pi

yields

4 = \frac{1}{9}r

, so

r = 36

.

Adım Adım Çözüm

1

Convert the swing angle of the pendulum from degrees to radians.

\theta = \frac{2\pi}{9}

radians

The arc length formula

s = r\theta

requires the angle

\theta

to be in radians.

2

Set up the arc length equation using

s = r\theta

, where

s = 8\pi

is the arc length and

r

is the length of the pendulum.

8\pi = r \left(\frac{2\pi}{9}\right)

The tip of the pendulum travels along a circular path whose radius is the length of the pendulum.

3

Solve the equation for the radius

r

.

r = 36

Isolating

r

by multiplying both sides by

\frac{9}{2\pi}

yields the length of the pendulum.

Anahtar Kavram

Converting angle measures between degrees and radians and applying the arc length formula.

Soru 116Soru →

In right triangle $XYZ$ , the measure of angle $XYZ$ is $90^\circ$ . Altitude $YW$ is drawn to hypotenuse $XZ$ , dividing the triangle into two smaller triangles, $\triangle XYW$ and $\triangle YZW$ . If the area of $\triangle XYW$ is $9$ and the area of $\triangle YZW$ is $36$ , what is the length of altitude $YW$ ?

3\sqrt{2}

6

12

18

Cevabı ve açıklamayı göster

Cevap: $6$

Cevap

The length of altitude

YW

is

6

.

Because altitude

YW

is perpendicular to hypotenuse

XZ

in right triangle

XYZ

, the two smaller triangles

\triangle XYW

and

\triangle YZW

are similar. The ratio of the area of

\triangle XYW

to the area of

\triangle YZW

is

9

to

36

, which simplifies to

1

to

4

. Since the ratio of the areas of similar triangles is the square of their linear scale factor, the ratio of their corresponding side lengths is

\sqrt{1/4} = 1/2

. Side

XW

corresponds to side

YW

, and side

YW

corresponds to side

WZ

, so

YW = 2 \cdot XW

. The area of

\triangle XYW

is given by

\frac{1}{2} \cdot XW \cdot YW = 9

. Substituting

YW = 2 \cdot XW

gives

\frac{1}{2} \cdot XW \cdot (2 \cdot XW) = 9

, which simplifies to

XW^2 = 9

, so

XW = 3

. Thus, the length of altitude

YW

is

2 \cdot 3 = 6

.

Adım Adım Çözüm

1

Set up the similarity relationship between the two smaller triangles.

\triangle XYW \sim \triangle YZW

because both are right triangles sharing an acute angle relationship with the main right triangle

\triangle XYZ

.

An altitude drawn to the hypotenuse of a right triangle divides it into two triangles that are similar to the original triangle and to each other.

2

Determine the scale factor between the similar triangles using their areas.

The ratio of the area of

\triangle XYW

to the area of

\triangle YZW

is

9/36 = 1/4

. The linear scale factor is the square root of the area ratio, which is

\sqrt{1/4} = 1/2

.

For similar figures, the ratio of their areas is the square of the ratio of their corresponding linear dimensions.

3

Express the relationship between the corresponding sides of the two triangles.

In

\triangle XYW

and

\triangle YZW

, the side

XW

corresponds to

YW

, and the side

YW

corresponds to

WZ

. Therefore,

YW = 2 \cdot XW

.

The linear scale factor of

1/2

means each side of the smaller triangle is half the length of the corresponding side of the larger triangle.

4

Use the area formula for

\triangle XYW

to solve for the length of

XW

.

\text{Area}(\triangle XYW) = \frac{1}{2} \cdot XW \cdot YW = 9 \implies \frac{1}{2} \cdot XW \cdot (2 \cdot XW) = 9 \implies XW^2 = 9 \implies XW = 3

.

Substituting the relationship between

YW

and

XW

into the area equation isolates the single variable

XW

.

5

Calculate the length of the altitude

YW

.

YW = 2 \cdot 3 = 6

.

Multiplying the length of

XW

by the scale factor of

2

yields the length of

YW

.

Anahtar Kavram

Similarity in right triangles and the geometric relationships of altitudes

Tahmini Süre:2m 30s

Soru 117Soru →

In right triangle $ABC$ , the measure of angle $C$ is $90^\circ$ . A point $D$ lies on side $AC$ such that $BD$ is the angle bisector of angle $ABC$ . If $BC = 28$ and $BD = 35$ , what is the length of segment $AD$ ?

Cevabı ve açıklamayı göster

Cevap: 75

Cevap

The length of segment

AD

is 75.

By applying the Pythagorean theorem to right triangle

BCD

, we find

CD = 21

. Then, by the angle bisector theorem, the ratio of

AB

to

BC

equals the ratio of

AD

to

CD

, which gives

AB = \frac{4}{3}AD

. Using the Pythagorean theorem on right triangle

ABC

, we solve

(\frac{4}{3}AD)^2 = 28^2 + (21 + AD)^2

to find the positive length

AD = 75

.

Adım Adım Çözüm

1

Find the length of

CD

using the Pythagorean theorem on right triangle

BCD

.

CD = 21

Since angle

C

is a right angle, triangle

BCD

is a right triangle with hypotenuse

BD

and leg

BC

.

2

Apply the angle bisector theorem to express

AB

in terms of

AD

.

AB = \frac{4}{3}AD

The angle bisector theorem states that

\frac{AD}{CD} = \frac{AB}{BC}

. Substituting

BC = 28

and

CD = 21

gives

\frac{AD}{21} = \frac{AB}{28}

.

3

Set up a quadratic equation using the Pythagorean theorem on right triangle

ABC

.

(\frac{4}{3}AD)^2 = 28^2 + (21 + AD)^2

In right triangle

ABC

, the hypotenuse is

AB

and the legs are

BC = 28

and

AC = CD + AD = 21 + AD

.

4

Solve the quadratic equation for

AD

.

AD = 75

Expanding and simplifying the equation yields

AD^2 - 54AD - 1575 = 0

, which factors as

(AD - 75)(AD + 21) = 0

. Since length must be positive,

AD = 75

.

Anahtar Kavram

Pythagorean Theorem and Angle Bisector Theorem

Alternatif Yöntem

Let

\theta = \angle DBC

. Since

BD

bisects angle

B

,

\angle ABC = 2\theta

. In right triangle

BCD

,

\cos\theta = \frac{BC}{BD} = \frac{28}{35} = \frac{4}{5}

. In right triangle

ABC

,

\cos(2\theta) = \frac{BC}{AB} = \frac{28}{AB}

. Using the double-angle identity

\cos(2\theta) = 2\cos^2\theta - 1

, we get

\frac{28}{AB} = 2(\frac{4}{5})^2 - 1 = \frac{7}{25}

, which gives

AB = 100

. Finally,

AC = \sqrt{AB^2 - BC^2} = \sqrt{100^2 - 28^2} = 96

, so

AD = AC - CD = 96 - 21 = 75

.

Tahmini Süre:2m 30s

Soru 118Soru →

A sector of a circle with center $O$ has an area of $54\pi$ . The perimeter of the sector is $36 + 6\pi$ . If the radius of the circle is an integer, what is the radius of the circle?

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

18

The area of a sector is given by

A = \frac{1}{2}rs

, where

r

is the radius and

s

is the arc length. Given

A = 54\pi

, we have

\frac{1}{2}rs = 54\pi

, which simplifies to

rs = 108\pi

, or

s = \frac{108\pi}{r}

. The perimeter of a sector is

P = 2r + s

. Given

P = 36 + 6\pi

, we can substitute

s

to get

2r + \frac{108\pi}{r} = 36 + 6\pi

. Multiplying the entire equation by

r

and rearranging terms yields the quadratic equation

2r^2 - (36 + 6\pi)r + 108\pi = 0

. Factoring by grouping gives

(2r - 6\pi)(r - 18) = 0

. This yields two potential solutions:

r = 3\pi

and

r = 18

. Since the radius is specified to be an integer, the correct answer is

18

.

Adım Adım Çözüm

1

Relate sector area and perimeter to radius and arc length

Area =

\frac{1}{2}rs = 54\pi

and Perimeter =

2r + s = 36 + 6\pi

, where

r

is the radius and

s

is the arc length of the sector.

The area of a sector with radius

r

and arc length

s

is given by

\frac{1}{2}rs

, and its perimeter consists of the two radii plus the arc length.

2

Express arc length in terms of radius

s = \frac{108\pi}{r}

Isolating

s

from the area equation allows for substitution into the perimeter equation.

3

Substitute and form a quadratic equation

2r + \frac{108\pi}{r} = 36 + 6\pi \implies 2r^2 - (36 + 6\pi)r + 108\pi = 0

Multiplying both sides of the equation by

r

eliminates the fraction and forms a standard quadratic equation.

4

Factor the quadratic equation

(2r - 6\pi)(r - 18) = 0

Grouping the terms as

(2r^2 - 36r) - (6\pi r - 108\pi) = 0

allows us to factor out

2r(r - 18) - 6\pi(r - 18) = 0

.

5

Identify the integer radius

r = 18

The two solutions to the equation are

r = 3\pi

and

r = 18

. Since the problem specifies that the radius is an integer, we select

18

.

Anahtar Kavram

Calculating sector area and perimeter using relationships between radius, arc length, and angle measures.

Soru 119Soru →

In the $xy$ -plane, the graph of the equation $x^2 + y^2 - 14x - 12y + q = 0$ is a circle that is tangent to the $y$ -axis, where $q$ is a constant. What is the value of $q$ ?

Cevabı ve açıklamayı göster

Cevap: 36

Cevap

36

To find the value of

q

, we convert the general form of the circle's equation into standard form

(x-h)^2 + (y-k)^2 = r^2

by completing the square. Grouping the variables gives

(x^2 - 14x) + (y^2 - 12y) = -q

. Adding

(14/2)^2 = 49

and

(12/2)^2 = 36

to both sides yields

(x - 7)^2 + (y - 6)^2 = 85 - q

. Thus, the center of the circle is

(7, 6)

and the radius squared is

r^2 = 85 - q

. Since the circle is tangent to the

y

-axis, its radius must equal the horizontal distance from its center to the

y

-axis, which is

7

. Therefore, the radius squared is

7^2 = 49

. Setting

85 - q = 49

and solving for

q

gives

q = 36

.

Adım Adım Çözüm

1

Rearrange and group the terms of the equation to prepare for completing the square.

(x^2 - 14x) + (y^2 - 12y) = -q

Grouping terms allows us to complete the square for the

x

and

y

variables independently.

2

Complete the square for both the

x

and

y

expressions by adding the square of half the coefficient of the linear term to both sides.

(x^2 - 14x + 49) + (y^2 - 12y + 36) = -q + 49 + 36

Adding

(14/2)^2 = 49

and

(12/2)^2 = 36

transforms the trinomials into perfect squares.

3

Factor the perfect square trinomials and simplify the right side of the equation.

(x - 7)^2 + (y - 6)^2 = 85 - q

This puts the equation in the standard form of a circle,

(x-h)^2 + (y-k)^2 = r^2

, identifying the center as

(7, 6)

and the radius squared as

r^2 = 85 - q

.

4

Determine the radius of the circle using the condition of tangency to the

y

-axis.

r = 7

A circle tangent to the

y

-axis has a radius equal to the absolute value of the x-coordinate of its center, which is

|7| = 7

.

5

Solve for the constant

q

by equating the two expressions for the radius squared.

85 - q = 49 \implies q = 36

Since

r = 7

, we have

r^2 = 49

. Setting

85 - q = 49

yields

q = 36

.

Anahtar Kavram

Completing the square to find the standard equation of a circle and applying coordinate geometry tangency conditions.

Soru 120Soru →

A digital designer is scaling two similar triangular logos. The smaller logo has a perimeter of $18$ centimeters and an area of $12$ square centimeters. If the larger logo has a perimeter of $54$ centimeters, what is the area, in square centimeters, of the larger logo?

36

72

108

324

Cevabı ve açıklamayı göster

Cevap: $108$

Cevap

108 square centimeters

The correct answer is

108

. Since the two triangular logos are similar, the ratio of their areas is the square of the ratio of their perimeters. The ratio of their perimeters is

\frac{54}{18} = 3

, meaning the larger logo has linear dimensions that are

3

times those of the smaller logo. Squaring this scale factor gives the area scaling factor, which is

3^2 = 9

. Therefore, the area of the larger logo is

12 \times 9 = 108

square centimeters.

Adım Adım Çözüm

1

Determine the linear scale factor between the two similar logos.

The linear scale factor

k

is

\frac{54}{18} = 3

.

For similar figures, the ratio of any corresponding linear dimensions (such as perimeters) is equal to the linear scale factor.

2

Find the area scaling factor.

The area scaling factor is

k^2 = 3^2 = 9

.

The ratio of the areas of two similar figures is equal to the square of their linear scale factor.

3

Calculate the area of the larger logo.

The area of the larger logo is

12 \times 9 = 108

square centimeters.

Multiplying the area of the smaller logo by the area scaling factor yields the area of the larger logo.

Anahtar Kavram

The ratio of the areas of two similar two-dimensional shapes is equal to the square of their linear scale factor.

Tahmini Süre:1m 30s

Geometry and Trigonometry

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

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Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

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Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

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Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

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Adım Adım Çözüm

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Adım Adım Çözüm

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Adım Adım Çözüm

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Cevap

Adım Adım Çözüm

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Cevap

Adım Adım Çözüm

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Adım Adım Çözüm

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Cevap

Adım Adım Çözüm

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Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

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Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram