Tüm alıştırma soruları

Soru 2581Soru →

The quadratic equation $1.5x^2 - kx + 6 = 0$ , where $k$ is a positive constant, has exactly one real solution. What is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

For the quadratic equation

1.5x^2 - kx + 6 = 0

to have exactly one real solution, the discriminant

b^2 - 4ac

must equal

0

. Substituting

a = 1.5

,

b = -k

, and

c = 6

gives

(-k)^2 - 4(1.5)(6) = k^2 - 36 = 0

, which yields

k^2 = 36

. Since

k

must be a positive constant,

k

must be

6

.

Adım Adım Çözüm

1

Identify the coefficients of the quadratic equation

1.5x^2 - kx + 6 = 0

.

a = 1.5

,

b = -k

, and

c = 6

To use the discriminant formula, we need to know the values of

a

,

b

, and

c

from the standard form

ax^2 + bx + c = 0

.

2

Set the discriminant equal to zero.

b^2 - 4ac = 0

A quadratic equation has exactly one real solution if and only if its discriminant is equal to zero.

3

Substitute the coefficients into the discriminant formula and simplify.

k^2 - 36 = 0

Substituting

a = 1.5

,

b = -k

, and

c = 6

into the formula gives

(-k)^2 - 4(1.5)(6) = k^2 - 36 = 0

.

4

Solve the equation for the positive constant

k

.

k = 6

Solving

k^2 = 36

gives

k = 6

or

k = -6

. Since the problem states that

k

is a positive constant, we choose

k = 6

.

Anahtar Kavram

Determining the number of real solutions of a quadratic equation using the discriminant

Soru 2582Soru →

In trapezoid $ABCD$ , the bases $AB$ and $CD$ are parallel. The diagonals $AC$ and $BD$ intersect at point $E$ . If the area of $\triangle ABE$ is $16\text{ cm}^2$ and the area of $\triangle CDE$ is $36\text{ cm}^2$ , what is the total area, in square centimeters, of trapezoid $ABCD$ ?

Cevabı ve açıklamayı göster

Cevap: 100

Cevap

The total area of trapezoid

ABCD

is

100\text{ cm}^2

.

Triangles

ABE

and

CDE

are similar with an area ratio of

16:36

, yielding a side length ratio of

2:3

. The adjacent triangles

\triangle ADE

and

\triangle BCE

each share an altitude with the base triangles, making their areas equal to the geometric mean of the base areas:

\sqrt{16 \times 36} = 24\text{ cm}^2

each. Summing the four regions gives

16 + 36 + 24 + 24 = 100\text{ cm}^2

.

Adım Adım Çözüm

1

Establish the similarity of triangles

ABE

and

CDE

.

\triangle ABE \sim \triangle CDE

Since bases

AB

and

CD

are parallel, alternate interior angles are congruent (

\angle EAB \cong \angle ECD

and

\angle EBA \cong \angle EDC

). By AA Similarity, the triangles are similar.

2

Calculate the linear scale factor between the similar triangles.

\frac{AE}{EC} = \frac{BE}{ED} = \sqrt{\frac{16}{36}} = \frac{2}{3}

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding side lengths.

3

Find the areas of the adjacent triangles

\triangle ADE

and

\triangle BCE

.

\text{Area}(\triangle ADE) = 24\text{ cm}^2

and

\text{Area}(\triangle BCE) = 24\text{ cm}^2

Triangles

ADE

and

CDE

share the same altitude from vertex

D

to diagonal

AC

, meaning the ratio of their areas is equal to the ratio of their bases:

\frac{\text{Area}(\triangle ADE)}{\text{Area}(\triangle CDE)} = \frac{AE}{EC} = \frac{2}{3} \implies \text{Area}(\triangle ADE) = \frac{2}{3} \times 36 = 24

. By similar logic,

\frac{\text{Area}(\triangle BCE)}{\text{Area}(\triangle ABE)} = \frac{EC}{AE} = \frac{3}{2} \implies \text{Area}(\triangle BCE) = \frac{3}{2} \times 16 = 24

.

4

Sum the areas of the four individual triangles.

\text{Area}(ABCD) = 16 + 36 + 24 + 24 = 100\text{ cm}^2

The total area of the trapezoid is the sum of the areas of the four non-overlapping triangles formed by its diagonals.

Anahtar Kavram

For any trapezoid with diagonals intersecting at

E

and parallel bases forming triangles of areas

A_1

and

A_2

, the other two triangles each have an area equal to

\sqrt{A_1 A_2}

, and the total area of the trapezoid is given by

(\sqrt{A_1} + \sqrt{A_2})^2

.

Soru 2583Soru →

In the standard $(x, y)$ coordinate plane, if a non-horizontal and non-vertical line has a slope of $m$ , then reflecting this line across the line $y = x$ produces a line with a slope of $\frac{1}{m}$ .

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: True

Cevap

True

Reflecting a point

(x, y)

across the line

y = x

swaps the coordinates to

(y, x)

. For any two points

(x_1, y_1)

and

(x_2, y_2)

on the original line, the slope is

m = \frac{y_2 - y_1}{x_2 - x_1}

. The corresponding points on the reflected line are

(y_1, x_1)

and

(y_2, x_2)

, yielding a slope of

m' = \frac{x_2 - x_1}{y_2 - y_1} = \frac{1}{m}

. Thus, the statement is true.

Adım Adım Çözüm

1

Identify the coordinate transformation representing reflection across the line

y = x

.

Reflecting any point

(x, y)

across the line

y = x

yields the point

(y, x)

.

By definition of reflection across the identity line, the

x

-coordinate and

y

-coordinate of each point are swapped.

2

Set up the slope formula for the original line using two distinct points

(x_1, y_1)

and

(x_2, y_2)

.

The slope of the original line is

m = \frac{y_2 - y_1}{x_2 - x_1}

.

The slope of a line is the ratio of vertical change (rise) to horizontal change (run).

3

Determine the slope of the reflected line using the transformed points

(y_1, x_1)

and

(y_2, x_2)

.

The slope of the reflected line is

m' = \frac{x_2 - x_1}{y_2 - y_1}

.

Applying the slope formula to the reflected points swaps the numerator and denominator.

4

Relate the slope of the reflected line,

m'

, to the slope of the original line,

m

.

Since

m' = \frac{x_2 - x_1}{y_2 - y_1} = \frac{1}{\frac{y_2 - y_1}{x_2 - x_1}} = \frac{1}{m}

, the slope of the reflected line is

\frac{1}{m}

.

Taking the reciprocal of the original slope fraction yields the reflected slope fraction, which is valid since the line is non-horizontal (

m \neq 0

) and non-vertical (

m

is defined).

Anahtar Kavram

Reflection of a line across

y = x

swaps the rise and run of the line, inverting its slope.

Soru 2584Soru →

An angle in standard position has a measure of $\frac{5\pi}{6}$ radians. What is the degree measure of this angle?

Cevabı ve açıklamayı göster

Cevap: 150

Cevap

The degree measure of the angle is 150.

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

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

. In this case, multiplying

\frac{5\pi}{6}

by

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

simplifies to

150^\circ

because the

\pi

terms cancel and

180

divided by

6

is

30

, which is then multiplied by

5

.

Adım Adım Çözüm

1

Multiply the given radian measure by the conversion factor

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

to convert from radians to degrees.

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

One full rotation is

360^\circ

, which is equal to

2\pi

radians. Therefore,

180^\circ = \pi

radians, yielding the conversion factor

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

.

2

Simplify the expression by canceling out common terms.

150^\circ

The common term

\pi

cancels out from the numerator and denominator, leaving

\frac{5 \times 180^\circ}{6} = 5 \times 30^\circ = 150^\circ

.

Anahtar Kavram

Converting radian measures to degree measures

Soru 2585Soru →

In the figure below, quadrilateral $ABCD$ is composed of two right triangles, $\triangle ABC$ and $\triangle ACD$ . The measure of $\angle ABC$ is $90^\circ$ , and the measure of $\angle ACD$ is $90^\circ$ . The side lengths are $AB = 3$ units and $BC = 4$ units. If the measure of $\angle CAD$ is $60^\circ$ , what is the length, in units, of segment $CD$ ?

\sqrt{21}

5\sqrt{3}

7\sqrt{3}

10

\sqrt{39}

Cevabı ve açıklamayı göster

Cevap: $5\sqrt{3}$

Cevap

The length of segment

CD

is

5\sqrt{3}

units.

The correct answer is the length of

5\sqrt{3}

units. By using the Pythagorean theorem on the first right triangle

\triangle ABC

, the length of the hypotenuse is

AC = \sqrt{3^2 + 4^2} = 5

. Since

\triangle ACD

is a 30-60-90 right triangle with a right angle at

C

and

\angle CAD = 60^\circ

, the side

AC

is the shorter leg (opposite the

30^\circ

angle). The length of the longer leg

CD

(opposite the

60^\circ

angle) is therefore

AC\sqrt{3} = 5\sqrt{3}

.

Adım Adım Çözüm

1

Use the Pythagorean theorem in right triangle

\triangle ABC

to find the length of the hypotenuse

AC

.

AC = 5

Since

\triangle ABC

is a right triangle with legs

AB = 3

and

BC = 4

, the hypotenuse is

AC = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5

.

2

Identify the type of right triangle for

\triangle ACD

.

\triangle ACD

is a 30-60-90 right triangle.

The triangle has a right angle at

C

(measure of

\angle ACD = 90^\circ

) and an acute angle at

A

(measure of

\angle CAD = 60^\circ

), which leaves the remaining angle

\angle ADC = 30^\circ

.

3

Apply the special right triangle ratios to find the length of leg

CD

.

CD = 5\sqrt{3}

In a 30-60-90 triangle, the leg opposite the

60^\circ

angle is

\sqrt{3}

times the leg opposite the

30^\circ

angle. Here,

AC = 5

is opposite the

30^\circ

angle, so the longer leg

CD = AC\sqrt{3} = 5\sqrt{3}

.

Anahtar Kavram

Using the Pythagorean theorem to find a shared side and then applying special right triangle ratios (30-60-90) to solve for an unknown length.

Alternatif Yöntem

Instead of using the special right triangle ratios directly, right triangle trigonometry can be applied:

\tan(60^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{CD}{AC}

. Since

\tan(60^\circ) = \sqrt{3}

and

AC = 5

, we have

\sqrt{3} = \frac{CD}{5}

, which yields

CD = 5\sqrt{3}

.

Tahmini Süre:1m 0s

Soru 2586Soru →

On the unit circle, a terminal ray starts at the positive $x$ -axis and rotates counterclockwise by $120^\circ$ , and then rotates counterclockwise by an additional $135^\circ$ . What is the radian measure of the final angle in standard position?

\frac{5\pi}{12}

\frac{5\pi}{7}

\frac{17\pi}{12}

\frac{17\pi}{24}

\frac{45900}{\pi}

Cevabı ve açıklamayı göster

Cevap: $\frac{17\pi}{12}$

Cevap

\frac{17\pi}{12}

The correct option is

\frac{17\pi}{12}

. First, the two counterclockwise rotations are added to find the total angle in standard position:

120^\circ + 135^\circ = 255^\circ

. To convert this angle into radians, it is multiplied by the conversion ratio

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

, giving

\frac{255\pi}{180}

. Dividing the numerator and denominator by 15 simplifies the fraction to

\frac{17\pi}{12}

.

Adım Adım Çözüm

1

Calculate the total counterclockwise rotation angle in degrees by summing the two individual angles.

120^\circ + 135^\circ = 255^\circ

Consecutive rotations in the same direction add together to find the final angle measure.

2

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

255^\circ \times \frac{\pi}{180^\circ} = \frac{255\pi}{180}

Since 180 degrees equals pi radians, multiplying by pi over 180 converts degrees to radians.

3

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor.

\frac{255\pi}{180} = \frac{17\pi}{12}

Simplifying the fraction by dividing both 255 and 180 by 15 yields the simplest form.

Anahtar Kavram

Unit Circle and Angle Measures (Radians and Degrees)

Soru 2587Soru →

In the standard $(x, y)$ coordinate plane, a triangle has vertices at $A(-3, -2)$ , $B(5, -2)$ , and $C(x, y)$ . If the area of the triangle is $24$ square units, which of the following could be the coordinates of vertex $C$ ?

(2, 1)

(2, 4)

(2, 6)

(2, 8)

(2, 10)

Cevabı ve açıklamayı göster

Cevap: $(2, 4)$

Cevap

(2, 4)

The correct answer is the coordinate

(2, 4)

. The base of the triangle is the segment

AB

, which is horizontal since both vertices lie on the line

y = -2

. The length of this base is

5 - (-3) = 8

units. Using the formula for the area of a triangle,

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

, we have

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

, which simplifies to

24 = 4 \times \text{height}

, giving a height of

6

units. Since the base lies along

y = -2

, the

y

-coordinate of the third vertex must be

6

units away from

-2

(either at

-2 + 6 = 4

or

-2 - 6 = -8

). The point

(2, 4)

satisfies this requirement.

Adım Adım Çözüm

1

Calculate the length of the base of the triangle.

The base segment

AB

is horizontal because both

A(-3, -2)

and

B(5, -2)

have the same

y

-coordinate of

-2

. The length of the base is the difference in their

x

-coordinates:

5 - (-3) = 8

units.

To use the area formula of a triangle, we first need to determine the length of one of its sides to act as the base.

2

Use the area formula of a triangle to find its height.

The formula for the area of a triangle is

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

. Substituting the given area of

24

and the base length of

8

:

24 = \frac{1}{2} \times 8 \times \text{height} \Rightarrow 24 = 4 \times \text{height} \Rightarrow \text{height} = 6

units.

Knowing the area and the base allows us to find the vertical distance (height) from the base to the third vertex.

3

Determine the possible

y

-coordinates of vertex

C

.

Since the base lies on the horizontal line

y = -2

, the

y

-coordinate of vertex

C

must be exactly

6

units away from

-2

vertically. This means

y = -2 + 6 = 4

or

y = -2 - 6 = -8

. Thus,

C

can be any point with a

y

-coordinate of

4

or

-8

.

The height represents the vertical distance from the horizontal base line to the third vertex.

4

Match the calculated

y

-coordinates with the given options.

Among the options, only the coordinate

(2, 4)

has a

y

-coordinate of

4

.

We identify the option that provides a valid set of coordinates for vertex

C

.

Anahtar Kavram

Calculating coordinates of a geometric figure's vertex using the area formula and coordinate distances.

Soru 2588Soru →

In the standard $(x, y)$ coordinate plane, a square has two opposite vertices at $(1, -3)$ and $(7, 5)$ . What is the area of the square, in square units?

Cevabı ve açıklamayı göster

Cevap: 50

Cevap

The area of the square is 50 square units.

The length of the diagonal of the square is found using the distance formula between the two opposite vertices:

d = \sqrt{(7 - 1)^2 + (5 - (-3))^2} = \sqrt{6^2 + 8^2} = 10

. The area of a square can be calculated using its diagonal length

d

with the formula

\text{Area} = \frac{d^2}{2}

. Substituting

d = 10

gives

\text{Area} = \frac{10^2}{2} = 50

square units.

Adım Adım Çözüm

1

Calculate the length of the diagonal of the square using the distance formula.

The diagonal length is

10

.

The distance between opposite vertices of a square represents the length of its diagonal.

2

Determine the area of the square using the diagonal length.

The area is

50

.

The area of a square with diagonal

d

is given by

\frac{d^2}{2}

.

Anahtar Kavram

Finding the area of a square on the coordinate plane using its diagonal.

Tahmini Süre:1m 30s

Soru 2589Soru →

Two lines in a coordinate plane, $T_1$ and $T_2$ , are perpendicular to each other. Line $T_1$ has the equation $3x - 5y = 15$ , and line $T_2$ has the equation $ax + 9y = 20$ , where $a$ is a constant. What is the value of $a$ ?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The value of the constant

a

is 15.

The correct answer is 15. The slope of the line

3x - 5y = 15

is

\frac{3}{5}

. Since the two lines are perpendicular, the slope of the second line must be the negative reciprocal of

\frac{3}{5}

, which is

-\frac{5}{3}

. The slope of the line

ax + 9y = 20

is

-\frac{a}{9}

. Equating the two slopes yields

-\frac{a}{9} = -\frac{5}{3}

, which simplifies to

a = 15

.

Adım Adım Çözüm

1

Find the slope of line

T_1

The slope of line

T_1

is

\frac{3}{5}

.

Writing the equation

3x - 5y = 15

in slope-intercept form

y = \frac{3}{5}x - 3

isolates the slope coefficient.

2

Find the perpendicular slope

The perpendicular slope is

-\frac{5}{3}

.

Perpendicular lines have slopes that are negative reciprocals of each other, so we invert the fraction and change the sign.

3

Determine the slope of line

T_2

in terms of

a

The slope of line

T_2

is

-\frac{a}{9}

.

Rewriting the equation

ax + 9y = 20

in slope-intercept form

y = -\frac{a}{9}x + \frac{20}{9}

isolates the slope coefficient.

4

Solve for

a

a = 15

Setting the slope of

T_2

equal to the perpendicular slope yields the equation

-\frac{a}{9} = -\frac{5}{3}

, which simplifies to

a = 15

.

Anahtar Kavram

The slopes of perpendicular lines are negative reciprocals of each other.

Soru 2590Soru →

In the standard $(x, y)$ coordinate plane, a line segment has endpoints $A(-2, 3)$ and $B(4, y)$ . If the slope of the line passing through $A$ and $B$ is $-\frac{2}{3}$ , what is the value of $y$ ?

-7

-6

-1

\frac{5}{3}

7

Cevabı ve açıklamayı göster

Cevap: -1

Cevap

The value of

y

is

-1

.

The correct option is the one showing

-1

. By applying the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

to the points

A(-2, 3)

and

B(4, y)

with slope

-\frac{2}{3}

, we set up the equation

\frac{y - 3}{4 - (-2)} = -\frac{2}{3}

. Simplifying the denominator yields

\frac{y - 3}{6} = -\frac{2}{3}

. Multiplying both sides by

6

gives

y - 3 = -4

, and solving for

y

yields

y = -1

.

Adım Adım Çözüm

1

Write down the slope formula and substitute the given values.

\frac{y - 3}{4 - (-2)} = -\frac{2}{3}

The slope

m

of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is defined as

m = \frac{y_2 - y_1}{x_2 - x_1}

.

2

Simplify the denominator on the left side of the equation.

\frac{y - 3}{6} = -\frac{2}{3}

Subtracting a negative number is equivalent to addition:

4 - (-2) = 4 + 2 = 6

.

3

Multiply both sides by 6 to isolate the numerator.

y - 3 = -4

Multiplying

-\frac{2}{3}

by

6

yields

-4

.

4

Solve for

y

by adding 3 to both sides of the equation.

y = -1

Isolating

y

gives

y = -4 + 3 = -1

.

Anahtar Kavram

Using the slope formula to find a missing coordinate

Tahmini Süre:1m 0s

Soru 2591Soru →

In parallelogram $ABCD$ , the ratio of the measure of angle $A$ to the measure of angle $B$ is $2:3$ . The measure of angle $C$ is $12$ degrees less than $3$ times the value of $x$ . What is the value of $x$ ?

28

40

-20

52

44

Cevabı ve açıklamayı göster

Cevap: 28

Cevap

28

The correct answer is 28. In a parallelogram, consecutive angles are supplementary, so the measure of angle

A

and the measure of angle

B

must sum to

180^\circ

. Based on the given ratio of

2:3

, the measure of angle

A

is calculated as

\frac{2}{5} \times 180^\circ = 72^\circ

. Since opposite angles in a parallelogram are equal, the measure of angle

C

is also

72^\circ

. The description '12 degrees less than 3 times the value of

x

' translates to the expression

3x - 12

. Equating this to

72^\circ

gives

3x - 12 = 72

, which simplifies to

3x = 84

and yields

x = 28

.

Adım Adım Çözüm

1

Determine the measures of angles

A

and

B

using their ratio and the properties of a parallelogram.

The measure of angle

A

is

72^\circ

and the measure of angle

B

is

108^\circ

.

Adjacent angles in a parallelogram are supplementary, meaning their sum is

180^\circ

. Given the ratio of the measure of angle

A

to the measure of angle

B

is

2:3

, we can express their measures as

2y

and

3y

respectively. Solving

2y + 3y = 180^\circ

gives

5y = 180^\circ \implies y = 36^\circ

. Therefore, the measure of angle

A

is

2(36^\circ) = 72^\circ

and the measure of angle

B

is

3(36^\circ) = 108^\circ

.

2

Relate the measure of angle

C

to the calculated angle measures.

The measure of angle

C

is

72^\circ

.

Opposite angles of a parallelogram are equal in measure. Since the measure of angle

A

is

72^\circ

, the measure of the opposite angle

C

must also be

72^\circ

.

3

Set up and solve the algebraic equation to find the value of

x

.

x = 28

We are given that the measure of angle

C

is

12

degrees less than

3

times the value of

x

, which translates to

3x - 12

. Setting this expression equal to

72^\circ

gives the equation

3x - 12 = 72

. Adding

12

to both sides yields

3x = 84

, and dividing by

3

gives

x = 28

.

Anahtar Kavram

Properties of Parallelograms

Tahmini Süre:1m 30s

Soru 2592Soru →

For an angle $\theta$ in standard position, match each description of its terminal side on the left with the corresponding coordinates of its intersection point on the unit circle on the right.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The terminal side in Quadrant II with reference angle

60^\circ

matches

(-\frac{1}{2}, \frac{\sqrt{3}}{2})

; in Quadrant III with reference angle

45^\circ

matches

(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})

; in Quadrant IV with reference angle

30^\circ

matches

(\frac{\sqrt{3}}{2}, -\frac{1}{2})

; in Quadrant III with reference angle

30^\circ

matches

(-\frac{\sqrt{3}}{2}, -\frac{1}{2})

.

Each terminal side is matched correctly to its coordinates by applying the quadrant signs to the trigonometric values of the reference angles. Quadrant II corresponds to

(-, +)

, Quadrant III corresponds to

(-, -)

, and Quadrant IV corresponds to

(+, -)

. Using standard unit circle coordinates, a

60^\circ

reference angle gives magnitudes of

(\frac{1}{2}, \frac{\sqrt{3}}{2})

, a

45^\circ

reference angle gives

(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})

, and a

30^\circ

reference angle gives

(\frac{\sqrt{3}}{2}, \frac{1}{2})

.

Adım Adım Çözüm

1

Determine the signs of the

x

- and

y

-coordinates based on the quadrant of the terminal side.

Quadrant II points have

(-, +)

coordinates; Quadrant III points have

(-, -)

coordinates; Quadrant IV points have

(+, -)

coordinates.

On the unit circle,

x = \cos\theta

and

y = \sin\theta

. Cosine is negative in Quadrants II and III, while sine is negative in Quadrants III and IV.

2

Find the absolute values of the coordinates using the reference angle.

A

30^\circ

reference angle corresponds to coordinates of magnitude

(\frac{\sqrt{3}}{2}, \frac{1}{2})

; a

45^\circ

reference angle corresponds to

(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})

; a

60^\circ

reference angle corresponds to

(\frac{1}{2}, \frac{\sqrt{3}}{2})

.

The reference angle determines the basic trigonometric values

\cos\theta_{\text{ref}}

and

\sin\theta_{\text{ref}}

.

3

Combine the quadrant signs and coordinate magnitudes to find the unique point.

Quadrant II with

60^\circ

reference angle is

(-\frac{1}{2}, \frac{\sqrt{3}}{2})

. Quadrant III with

45^\circ

reference angle is

(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})

. Quadrant IV with

30^\circ

reference angle is

(\frac{\sqrt{3}}{2}, -\frac{1}{2})

. Quadrant III with

30^\circ

reference angle is

(-\frac{\sqrt{3}}{2}, -\frac{1}{2})

.

Applying the quadrant signs from Step 1 to the magnitude values from Step 2 yields the exact coordinates on the unit circle.

Anahtar Kavram

Coordinates of points on the unit circle are given by

(\cos\theta, \sin\theta)

, where the magnitude is determined by the reference angle and the signs are determined by the quadrant of the angle.

Soru 2593Soru →

A convex hexagon has interior angles with measures of $2x^\circ$ , $(3x - 15)^\circ$ , $(2x + 25)^\circ$ , $(3x + 10)^\circ$ , $(4x - 50)^\circ$ , and $(x + 30)^\circ$ . What is the degree measure of the smallest interior angle of this hexagon?

Cevabı ve açıklamayı göster

Cevap: 78

Cevap

The degree measure of the smallest interior angle is 78.

The sum of the interior angles of a hexagon is calculated as

(6 - 2) \times 180^\circ = 720^\circ

. Adding the algebraic expressions for the six angles yields

15x = 720

, which gives

x = 48

. Substituting

x = 48

back into the expressions gives the angle measures of

96^\circ

,

129^\circ

,

121^\circ

,

154^\circ

,

142^\circ

, and

78^\circ

. The smallest angle is

78^\circ

.

Adım Adım Çözüm

1

Calculate the sum of the interior angles of the hexagon.

Sum of interior angles = 720 degrees

The sum of the interior angles of any convex polygon with

n

sides is given by

(n - 2) \times 180^\circ

. For a hexagon (

n = 6

), the sum is

(6 - 2) \times 180^\circ = 720^\circ

.

2

Set up an algebraic equation by summing all the given angle expressions and equating them to 720.

2x + (3x - 15) + (2x + 25) + (3x + 10) + (4x - 50) + (x + 30) = 720

The sum of the actual measures of the interior angles must equal the calculated sum of 720 degrees.

3

Combine like terms to simplify the equation.

15x = 720

Grouping the

x

terms (

2x + 3x + 2x + 3x + 4x + x = 15x

) and the constant terms (

-15 + 25 + 10 - 50 + 30 = 0

) simplifies the expression.

4

Solve for

x

.

x = 48

Dividing both sides of the equation

15x = 720

by 15 isolates the variable

x

.

5

Substitute the value of

x

back into the angle expressions to identify the smallest angle.

The angles are

96^\circ

,

129^\circ

,

121^\circ

,

154^\circ

,

142^\circ

, and

78^\circ

. The smallest measure is

78^\circ

.

Evaluating each expression at

x = 48

determines the actual angle measures, from which the smallest can be chosen. Evaluating

(x + 30)^\circ

gives

48 + 30 = 78^\circ

, which is the minimum value.

Anahtar Kavram

The sum of the interior angles of a convex

n

-sided polygon is

(n - 2) \times 180^\circ

. Setting up and solving linear algebraic equations is required to determine unknown angle measures.

Tahmini Süre:1m 30s

Soru 2594Soru →

A circular search-and-rescue radar zone centered at a local station is modeled by the equation $x^2 + y^2 = 25$ in the standard $(x, y)$ coordinate plane, where coordinates are measured in miles. A rescue helicopter flies along a straight path modeled by the line $y = 2x - 5$ . What is the distance, in miles, the helicopter travels through the radar zone?

2\sqrt{5}

4\sqrt{5}

8

10

10\sqrt{2}

Cevabı ve açıklamayı göster

Cevap: $4\sqrt{5}$

Cevap

The distance the helicopter travels through the radar zone is

4\sqrt{5}

miles.

To find the distance the helicopter travels through the radar zone, we must determine the distance between the two points of intersection of the circular boundary

x^2 + y^2 = 25

and the line

y = 2x - 5

. Substituting the expression for

y

into the circular equation yields

x^2 + (2x-5)^2 = 25

. Expanding the binomial correctly gives

x^2 + 4x^2 - 20x + 25 = 25

, which simplifies to

5x^2 - 20x = 0

. Factoring this equation as

5x(x-4) = 0

gives

x = 0

and

x = 4

. Substituting these values back into the linear equation yields the points of intersection

(0, -5)

and

(4, 3)

. The distance between these two points is

\sqrt{(4-0)^2 + (3 - (-5))^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}

miles.

Adım Adım Çözüm

1

Substitute the linear equation

y = 2x - 5

into the circular equation

x^2 + y^2 = 25

.

x^2 + (2x - 5)^2 = 25

To find the coordinates of the intersection points where the helicopter's path meets the boundary of the radar zone.

2

Expand the binomial

(2x - 5)^2

and simplify the quadratic equation.

x^2 + 4x^2 - 20x + 25 = 25 \implies 5x^2 - 20x = 0

To collect like terms and put the equation in a solvable quadratic form.

3

Factor the quadratic equation

5x^2 - 20x = 0

to solve for

x

.

5x(x - 4) = 0 \implies x = 0 \text{ or } x = 4

To determine the

x

-coordinates of the two intersection points.

4

Determine the corresponding

y

-coordinates by substituting the

x

-values into the linear equation

y = 2x - 5

.

For

x = 0

:

y = 2(0) - 5 = -5 \implies (0, -5)

. For

x = 4

:

y = 2(4) - 5 = 3 \implies (4, 3)

.

To obtain the exact coordinate pairs for the entry and exit points.

5

Use the distance formula to calculate the distance between the two points

(0, -5)

and

(4, 3)

.

d = \sqrt{(4 - 0)^2 + (3 - (-5))^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}

To compute the straight-line distance traveled by the helicopter through the radar zone.

Anahtar Kavram

Solving systems of linear and circular equations by substitution and finding the distance between their intersection points.

Soru 2595Soru →

What is the greatest integer value of $x$ that satisfies the inequality $\frac{7 - 5x}{3} - 4 \ge 2(x + 6)$ ?

Cevabı ve açıklamayı göster

Cevap: -4

Cevap

The greatest integer value of

x

that satisfies the inequality is

-4

.

To find the greatest integer value of

x

that satisfies the inequality, solve the inequality algebraically. First, clear the fraction by multiplying all terms by 3:

7 - 5x - 12 \ge 6(x + 6)

. Simplify the left side to

-5x - 5

and distribute the right side to get

6x + 36

. Move variables to the left side by subtracting

6x

to get

-11x - 5 \ge 36

. Add 5 to both sides to get

-11x \ge 41

. Finally, divide by

-11

and reverse the inequality sign, yielding

x \le -\frac{41}{11}

, which is approximately

x \le -3.73

. The greatest integer less than or equal to

-3.73

is

-4

.

Adım Adım Çözüm

1

Multiply the entire inequality by 3 to clear the fraction

7 - 5x - 12 \ge 6(x + 6)

Multiplying by the common denominator eliminates the fraction and simplifies further algebraic steps.

2

Simplify the left side and distribute the right side

-5x - 5 \ge 6x + 36

Combines constant terms on the left side and expands the parentheses on the right side.

3

Subtract

6x

from both sides

-11x - 5 \ge 36

Gathers all variable terms on the left side of the inequality.

4

Add 5 to both sides

-11x \ge 41

Isolates the variable term by moving the constant to the right side.

5

Divide by -11 and reverse the inequality sign

x \le -\frac{41}{11}

Isolates the variable

x

. Reversing the inequality sign is required when multiplying or dividing both sides by a negative number.

6

Determine the greatest integer satisfying the inequality

x \le -3.73

, so the greatest integer is

-4

Since

-\frac{41}{11} \approx -3.73

, the largest integer that is less than or equal to this value is

-4

.

Anahtar Kavram

Solving linear inequalities, applying the sign-flip rule when dividing by a negative number, and identifying integer boundaries.

Soru 2596Soru →

A pendulum of a certain length swings back and forth such that the tip of the pendulum traces an arc of length $6\pi$ inches. If the length of the pendulum is increased by $4$ inches and it swings through the same central angle, the tip of the pendulum traces an arc of length $8\pi$ inches. What is the area, in square inches, of the circular sector swept out by the original pendulum?

12\pi

18\pi

36\pi

72\pi

144\pi

Cevabı ve açıklamayı göster

Cevap: $36\pi$

Cevap

The correct answer is

36\pi

square inches.

The correct answer is

36\pi

square inches. By using the arc length formula

s = r\theta

where

\theta

is in radians, we establish the system of equations

r\theta = 6\pi

and

(r+4)\theta = 8\pi

. Subtracting the first equation from the second gives

4\theta = 2\pi

, which simplifies to

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

. Substituting this back into the first equation yields

r(\frac{\pi}{2}) = 6\pi

, so the original radius

r = 12

. Finally, we calculate the area of the original sector using the formula

A = \frac{1}{2}r^2\theta = \frac{1}{2}(12)^2(\frac{\pi}{2}) = 36\pi

.

Adım Adım Çözüm

1

Write down the arc length equations for both pendulums using the formula

s = r\theta

, where

r

is the length of the pendulum (radius) and

\theta

is the central angle in radians.

For the original pendulum:

r\theta = 6\pi

. For the extended pendulum:

(r+4)\theta = 8\pi

.

To establish the mathematical relationships between the given arc lengths, the pendulum lengths, and the central angle.

2

Solve the system of equations for the central angle

\theta

.

Distributing the second equation gives

r\theta + 4\theta = 8\pi

. Substituting

r\theta = 6\pi

into this yields

6\pi + 4\theta = 8\pi \implies 4\theta = 2\pi \implies \theta = \frac{\pi}{2}

radians.

To find the constant central angle of the pendulum's swing.

3

Substitute

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

back into the first equation to solve for the original radius

r

.

r\left(\frac{\pi}{2}\right) = 6\pi \implies r = 12

inches.

To find the length of the original pendulum, which serves as the radius of the sector.

4

Calculate the area of the sector swept out by the original pendulum using the formula

A = \frac{1}{2}r^2\theta

.

A = \frac{1}{2}(12)^2\left(\frac{\pi}{2}\right) = \frac{1}{2}(144)\left(\frac{\pi}{2}\right) = 36\pi

square inches.

To find the final area of the sector as requested by the question.

Anahtar Kavram

Calculating sector area using arc length relationships to determine radius and angle.

Alternatif Yöntem

Instead of solving for

\theta

first, one can note that the ratio of the arc lengths is equal to the ratio of the radii because the central angle is constant:

\frac{r}{r+4} = \frac{6\pi}{8\pi} = \frac{3}{4}

. Solving for

r

gives

4r = 3r + 12 \implies r = 12

. Since the arc length of the original sector is

s = 6\pi

, we can use the sector area formula

A = \frac{1}{2}rs = \frac{1}{2}(12)(6\pi) = 36\pi

square inches.

Tahmini Süre:3m 0s

Soru 2597Soru →

A hiker starts at a trailhead, point $A$ , and walks $9$ miles due east, then $12$ miles due north to reach a campsite, point $C$ . A lookout tower, point $T$ , is located due west of the campsite $C$ . The straight-line distance from the starting point $A$ to the tower $T$ is $13$ miles. If the tower $T$ is located east of the north-south line passing through point $A$ , what is the distance, in miles, between the campsite $C$ and the tower $T$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The distance between the campsite and the tower is 4 miles.

The correct answer is 4 miles. By modeling the hiker's path, the vertical height of both the campsite and the tower is 12 miles north of the starting point's east-west line. A right triangle is formed by the trailhead, the projection of the tower onto the east-west axis, and the tower itself. The hypotenuse is 13 miles and the vertical leg is 12 miles. By the Pythagorean theorem, the horizontal leg is 5 miles. Since the tower is east of the trailhead's north-south line, it is 5 miles east of the trailhead. The campsite is 9 miles east of the trailhead. The horizontal distance between the campsite and the tower is the difference: 9 - 5 = 4 miles.

Adım Adım Çözüm

1

Determine the vertical height of the campsite and the tower.

The campsite

C

is

12

miles north of the trailhead

A

's east-west line. Since the tower

T

is located due west of

C

, it lies on the same horizontal line. Therefore, the vertical distance from the east-west line to the tower

T

is also

12

miles.

Points on the same due east-west line share the same vertical offset (latitude) from the reference axis.

2

Use the Pythagorean theorem to find the horizontal distance from the trailhead to the tower.

Let

D

be the point on the east-west line directly below the tower

T

. A right triangle

\triangle ADT

is formed where the vertical leg

DT = 12

miles, the hypotenuse

AT = 13

miles, and the horizontal leg is

AD

. Using the Pythagorean theorem:

AD^2 + DT^2 = AT^2 \implies AD^2 + 12^2 = 13^2 \implies AD^2 + 144 = 169 \implies AD^2 = 25 \implies AD = 5

miles.

Calculating the horizontal offset of the tower from the trailhead's north-south line.

3

Calculate the horizontal distance between the campsite and the tower.

The campsite

C

is

9

miles east of the trailhead

A

's north-south line, and the tower

T

is

5

miles east of it. The horizontal distance between them is the difference:

9 - 5 = 4

miles.

Since both points have the same vertical coordinate, the direct distance is simply the difference in their horizontal coordinates.

Anahtar Kavram

Using the Pythagorean theorem to solve multi-step geometry problems on a coordinate-like plane.

Tahmini Süre:1m 30s

Soru 2598Soru →

A particle starts at the point $(1, 0)$ on the unit circle in the standard coordinate plane. It first travels along the circle counterclockwise by $\frac{11\pi}{4}$ radians, then travels clockwise by $120^\circ$ , and finally travels counterclockwise by $\frac{\pi}{4}$ radians. Which of the following ordered pairs represents the coordinates of the particle's final position?

\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)

\left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right)

\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)

(1, 0)

\left( \frac{\sqrt{3}}{2}, -\frac{1}{2} \right)

Cevabı ve açıklamayı göster

Cevap: $\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)$

Cevap

\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)

The correct answer is the coordinate pair representing a net rotation of

\frac{\pi}{3}

radians. Since the particle starts at

(1, 0)

, its coordinates after a net rotation of

\frac{\pi}{3}

radians are given by

(\cos(\frac{\pi}{3}), \sin(\frac{\pi}{3}))

, which evaluates to

\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)

.

Adım Adım Çözüm

1

Determine the sign and radian value of each rotation.

The first rotation (counterclockwise) is

+\frac{11\pi}{4}

radians. The second rotation (clockwise) is negative:

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

radians. The third rotation (counterclockwise) is

+\frac{\pi}{4}

radians.

Standard orientation defines counterclockwise rotation as positive and clockwise rotation as negative. Angles must be in the same unit (radians) to be combined.

2

Calculate the net angle of rotation by summing the signed values.

\theta_{net} = \frac{11\pi}{4} - \frac{2\pi}{3} + \frac{\pi}{4} = \left(\frac{11\pi}{4} + \frac{\pi}{4}\right) - \frac{2\pi}{3} = 3\pi - \frac{2\pi}{3} = \frac{7\pi}{3}

radians.

Grouping terms with the same denominator simplifies the fraction arithmetic.

3

Find the coterminal angle of

\frac{7\pi}{3}

in the interval

[0, 2\pi)

.

\frac{7\pi}{3} = 2\pi + \frac{\pi}{3}

, which is coterminal to

\frac{\pi}{3}

radians.

Subtracting integer multiples of

2\pi

(full revolutions) gives the standard position of the terminal ray.

4

Evaluate the coordinates of the point on the unit circle at

\frac{\pi}{3}

radians.

(x, y) = \left(\cos\left(\frac{\pi}{3}\right), \sin\left(\frac{\pi}{3}\right)\right) = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)

.

For any angle

\theta

on the unit circle, the coordinates are given by

(\cos\theta, \sin\theta)

.

Anahtar Kavram

Unit circle coordinates, angle conversion, and coterminal angles

Tahmini Süre:2m 0s

Soru 2599Soru →

In the standard $(x, y)$ coordinate plane, a line passes through the points $(-\frac{1}{2}, \frac{2}{3})$ and $(\frac{3}{4}, -\frac{1}{2})$ . What is the slope of this line?

\frac{14}{15}

-\frac{15}{14}

-\frac{9}{10}

-\frac{14}{15}

-\frac{14}{3}

Cevabı ve açıklamayı göster

Cevap: $-\frac{14}{15}$

Cevap

The slope of the line is

-\frac{14}{15}

.

The slope of the line is found by using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

. Substituting the given points

(-\frac{1}{2}, \frac{2}{3})

and

(\frac{3}{4}, -\frac{1}{2})

into the formula, we find the change in

y

is

-\frac{7}{6}

and the change in

x

is

\frac{5}{4}

. Dividing the change in

y

by the change in

x

yields

-\frac{14}{15}

.

Adım Adım Çözüm

1

Identify the coordinates of the two points as

(x_1, y_1) = (-\frac{1}{2}, \frac{2}{3})

and

(x_2, y_2) = (\frac{3}{4}, -\frac{1}{2})

.

x_1 = -\frac{1}{2}

,

y_1 = \frac{2}{3}

,

x_2 = \frac{3}{4}

, and

y_2 = -\frac{1}{2}

To use the slope formula, we must first map the given points to their respective coordinate variables.

2

Apply the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

to calculate the change in

y

(rise) and the change in

x

(run).

Rise =

-\frac{1}{2} - \frac{2}{3} = -\frac{7}{6}

; Run = \frac{3}{4} - (-\frac{1}{2}) = \frac{5}{4}

The slope is the ratio of the vertical change to the horizontal change.

3

Divide the change in

y

by the change in

x

and simplify the resulting fraction.

m = \frac{-\frac{7}{6}}{\frac{5}{4}} = -\frac{7}{6} \times \frac{4}{5} = -\frac{28}{30} = -\frac{14}{15}

Dividing fractions is performed by multiplying the numerator fraction by the reciprocal of the denominator fraction.

Anahtar Kavram

Calculating the slope of a line between two points containing positive and negative fractional coordinates.

Soru 2600Soru →

For an angle in standard position on the unit circle, match each rotation scenario on the left with its corresponding terminal angle and location on the right.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

Matching the scenarios: the wheel rotation matches

\frac{3\pi}{2}

radians on the negative

y

-axis; the combined rotation of

\frac{5\pi}{3}

and

-450^\circ

matches

\frac{7\pi}{6}

radians in Quadrant III; the clockwise travel of

\frac{11\pi}{4}

units matches

\frac{5\pi}{4}

radians in Quadrant III; the combined rotation of

-150^\circ

and

\frac{11\pi}{6}

matches

\pi

radians on the negative

x

-axis.

Each scenario is correctly matched by converting all angular values to radians, determining the net rotation direction (positive for counterclockwise, negative for clockwise), and finding the coterminal angle in the range

[0, 2\pi)

to locate the terminal side.

Adım Adım Çözüm

1

Convert the rotation from revolutions to radians for the first scenario.

3.75 \text{ revolutions} \times 2\pi \text{ radians/revolution} = 7.5\pi

radians. Subtract

3

full rotations (

6\pi

radians) to find the coterminal angle in

[0, 2\pi)

:

7.5\pi - 6\pi = 1.5\pi = \frac{3\pi}{2}

radians. This lies on the negative

y

-axis.

One full revolution corresponds to

2\pi

radians, and subtracting multiples of

2\pi

yields the coterminal position.

2

Calculate the net angle in radians for the second scenario.

Convert

450^\circ

to radians:

-450^\circ \times \frac{\pi}{180^\circ} = -\frac{5\pi}{2}

radians (negative due to clockwise direction). Net angle is

\frac{5\pi}{3} - \frac{5\pi}{2} = -\frac{5\pi}{6}

radians. Find the positive coterminal angle:

-\frac{5\pi}{6} + 2\pi = \frac{7\pi}{6}

radians. This lies in Quadrant III.

Converting all angles to radians with correct sign conventions allows addition to find the net angle.

3

Relate arc length to angle measure on the unit circle for the third scenario.

On a circle with

r = 1

, the arc length

s = \frac{11\pi}{4}

corresponds to a rotation of

\frac{11\pi}{4}

radians. Clockwise direction makes it

-\frac{11\pi}{4}

radians. Find the coterminal angle in

[0, 2\pi)

:

-\frac{11\pi}{4} + 4\pi = \frac{5\pi}{4}

radians. This lies in Quadrant III.

The arc length formula

s = r\theta

simplifies to

s = \theta

on the unit circle, and clockwise motion represents a negative angle.

4

Compute the net angle in radians for the fourth scenario.

Convert

-150^\circ

to radians:

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

radians. Net angle is

-\frac{5\pi}{6} + \frac{11\pi}{6} = \frac{6\pi}{6} = \pi

radians. This lies on the negative

x

-axis.

Converting degrees to radians enables direct fraction addition to determine the final terminal position.

Anahtar Kavram

Calculating coterminal angles and conversions between degrees, radians, and revolutions on the unit circle.

Tahmini Süre:2m 30s

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Alternatif Yöntem

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Adım Adım Çözüm

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Adım Adım Çözüm

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Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap