Tüm alıştırma soruları

Soru 2541Soru →

A convex polygon has $n$ sides. The interior angles of the polygon form an arithmetic progression with a common difference of $d^\circ$ , where $d$ is a positive integer. If the smallest interior angle of the polygon measures $100^\circ$ , what is the maximum possible value of $n$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The maximum possible value of

n

is 8.

The correct answer is 8 because we set the sum of the interior angles

(n-2) \times 180^\circ

equal to the sum of the arithmetic progression

\frac{n}{2}[200 + (n-1)d]

. Solving for

(n-1)d

gives

(n-1)d = 160 - \frac{720}{n}

. Since the polygon is convex, the largest angle

100 + (n-1)d

must be strictly less than

180^\circ

, which means

(n-1)d < 80^\circ

. Substituting the expression yields

160 - \frac{720}{n} < 80

, which simplifies to

n < 9

. Since

n

must be an integer, the maximum possible value of

n

is 8. For

n = 8

, the common difference

d = 10

is a positive integer, satisfying all conditions.

Adım Adım Çözüm

1

Express the sum of the interior angles using the polygon angle sum formula and the arithmetic progression formula.

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

. The sum of the angles in arithmetic progression with first term

100^\circ

and common difference

d^\circ

is

\frac{n}{2}[2(100) + (n-1)d] = 100n + \frac{n(n-1)d}{2}

.

This sets up the system relating the geometry of the polygon to the algebraic progression of its angles.

2

Equate the two expressions and solve for the quantity

(n-1)d

.

100n + \frac{n(n-1)d}{2} = 180n - 360 \implies \frac{n(n-1)d}{2} = 80n - 360 \implies (n-1)d = 160 - \frac{720}{n}

.

This isolates the quantity

(n-1)d

, which represents the difference between the largest and smallest angles.

3

Apply the convexity constraint that every interior angle must be strictly less than

180^\circ

.

The largest angle is the last term of the progression:

100 + (n-1)d

. For the polygon to be convex, this angle must be strictly less than

180^\circ

. Therefore,

100 + (n-1)d < 180 \implies (n-1)d < 80

. Substituting

(n-1)d = 160 - \frac{720}{n}

gives

160 - \frac{720}{n} < 80 \implies 80 < \frac{720}{n} \implies n < 9

.

A convex polygon cannot have any interior angles greater than or equal to

180^\circ

.

4

Identify the maximum integer value of

n

and verify that a positive integer common difference

d

exists.

Since

n < 9

and

n

must be an integer, the maximum possible value is

n = 8

. For

n = 8

, we calculate

(8-1)d = 160 - \frac{720}{8} \implies 7d = 70 \implies d = 10

. Since

d = 10

is a positive integer, the solution is valid.

This ensures the result satisfies all constraints, including that the common difference is a positive integer.

Anahtar Kavram

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, and all interior angles of a convex polygon must be strictly less than

180^\circ

.

Soru 2542Soru →

In the standard $(x, y)$ coordinate system, a line given by the equation $y = 2x + 4$ intersects a parabola given by the equation $y = x^2 - 2x - 1$ at exactly two points. What is the sum of the $y$ -coordinates of these two points of intersection?

Cevabı ve açıklamayı göster

Cevap: 16

Cevap

16

The system of equations is solved by setting the equations equal to each other, yielding the quadratic equation

x^2 - 4x - 5 = 0

. Solving for

x

gives

x = 5

and

x = -1

. Substituting these values into the linear equation gives the y-coordinates

14

and

2

. The sum of these y-coordinates is

14 + 2 = 16

.

Adım Adım Çözüm

1

Equate the linear and quadratic equations to find the x-values of the intersection points.

x^2 - 2x - 1 = 2x + 4

At the points of intersection, the y-values of both equations must be equal.

2

Rearrange the equation to standard quadratic form.

x^2 - 4x - 5 = 0

Grouping all terms on one side allows the quadratic equation to be solved.

3

Factor the quadratic equation to find the x-coordinates.

(x - 5)(x + 1) = 0

, so

x = 5

or

x = -1

The numbers that multiply to

-5

and add up to

-4

are

-5

and

1

.

4

Substitute the x-coordinates into the linear equation to find the y-coordinates.

For

x = 5

:

y = 2(5) + 4 = 14

. For

x = -1

:

y = 2(-1) + 4 = 2

.

Evaluating the linear equation is simpler than evaluating the quadratic equation.

5

Find the sum of the y-coordinates.

14 + 2 = 16

The question asks for the sum of the y-coordinates of the two points of intersection.

Anahtar Kavram

Solving a system of linear and quadratic equations by substitution.

Alternatif Yöntem

Instead of solving for the individual intersection points, Vieta's formulas can be applied. The x-coordinates satisfy

x^2 - 4x - 5 = 0

, so their sum is

x_1 + x_2 = 4

. Since the points lie on the line

y = 2x + 4

, the sum of the y-coordinates is

y_1 + y_2 = (2x_1 + 4) + (2x_2 + 4) = 2(x_1 + x_2) + 8 = 2(4) + 8 = 16

.

Tahmini Süre:1m 30s

Soru 2543Soru →

A parallelogram with vertices $P(-3, -2)$ , $Q(5, -2)$ , $R(8, 2)$ , and $S(0, 2)$ is plotted on a standard coordinate grid. What is the perimeter of parallelogram $PQRS$ ?

16 + 2\sqrt{7}

26

30

32

34

Cevabı ve açıklamayı göster

Cevap: 26

Cevap

The correct answer is 26, which is the perimeter of the parallelogram.

The horizontal base segments have a length of

5 - (-3) = 8

. The slanted side has a length of

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

. Summing the four sides gives a perimeter of

2(8) + 2(5) = 26

.

Adım Adım Çözüm

1

Calculate the length of the horizontal base PQ using the x-coordinates of P(-3, -2) and Q(5, -2).

PQ = 5 - (-3) = 8

Horizontal segments on the coordinate plane have a length equal to the difference in their x-coordinates.

2

Apply the distance formula to find the length of the slanted side PS with vertices P(-3, -2) and S(0, 2).

PS = \sqrt{(0 - (-3))^2 + (2 - (-2))^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5

The distance formula determines the straight-line distance between two coordinates.

3

Calculate the total perimeter of parallelogram PQRS by summing the lengths of all four sides.

\text{Perimeter} = 2 \times \text{base} + 2 \times \text{slanted side} = 2(8) + 2(5) = 16 + 10 = 26

The perimeter of a parallelogram is the sum of all its outer boundary lengths.

Anahtar Kavram

Calculating the perimeter of geometric figures on the coordinate plane by determining horizontal distances and applying the distance formula for skewed lines.

Alternatif Yöntem

Instead of using the distance formula, you can sketch a right triangle for the slanted side PS. The horizontal leg has length 3 and the vertical leg has length 4. Recognizing this as a standard 3-4-5 right triangle, the hypotenuse (the slanted side) must be 5.

Tahmini Süre:1m 30s

Soru 2544Soru →

On a map of a city, Oak Street is represented by a straight line with the equation $3x - 4y = 12$ . A new road, Pine Street, is planned to be perpendicular to Oak Street and will pass through a park located at the coordinates $(2, -1)$ . Which of the following is an equation that represents Pine Street?

3x + 4y = 2

4x - 3y = 11

4x + 3y = 5

4x + 3y = 7

4x + 3y = -9

Cevabı ve açıklamayı göster

Cevap: $4x + 3y = 5$

Cevap

4x + 3y = 5

The slope of Oak Street is

\frac{3}{4}

. The perpendicular line representing Pine Street must have a slope that is the negative reciprocal of

\frac{3}{4}

, which is

-\frac{4}{3}

. Using the point-slope form with the point

(2, -1)

, we get

y - (-1) = -\frac{4}{3}(x - 2)

. Simplifying this yields

y + 1 = -\frac{4}{3}x + \frac{8}{3}

, which becomes

y = -\frac{4}{3}x + \frac{5}{3}

. Multiplying the entire equation by 3 and moving the

x

term to the left side gives the standard form equation

4x + 3y = 5

.

Adım Adım Çözüm

1

Find the slope of Oak Street by converting its equation to slope-intercept form (

y = mx + b

).

The equation

3x - 4y = 12

becomes

-4y = -3x + 12

, which simplifies to

y = \frac{3}{4}x - 3

. The slope of Oak Street is

\frac{3}{4}

.

To find the slope of a perpendicular line, we must first determine the slope of the original line.

2

Determine the perpendicular slope of Pine Street by taking the negative reciprocal of Oak Street's slope.

The negative reciprocal of

\frac{3}{4}

is

-\frac{4}{3}

.

Perpendicular lines have slopes that are negative reciprocals of each other.

3

Use the point-slope formula

y - y_1 = m(x - x_1)

with the point

(2, -1)

and the perpendicular slope

-\frac{4}{3}

to find the equation of Pine Street.

Substituting the values gives

y - (-1) = -\frac{4}{3}(x - 2)

, which simplifies to

y + 1 = -\frac{4}{3}x + \frac{8}{3}

.

The point-slope formula allows us to write the equation of a line given its slope and a point it passes through.

4

Convert the equation to standard form (

Ax + By = C

).

Subtract 1 from both sides:

y = -\frac{4}{3}x + \frac{8}{3} - \frac{3}{3} \Rightarrow y = -\frac{4}{3}x + \frac{5}{3}

. Multiply the entire equation by 3:

3y = -4x + 5

. Add

4x

to both sides:

4x + 3y = 5

.

Converting to standard form matches the format of the options provided in the question.

Anahtar Kavram

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other (

m_1 \cdot m_2 = -1

). Once the perpendicular slope is found, the point-slope formula can be used to write the equation of the line passing through a given point.

Soru 2545Soru →

An irregular convex octagon has five interior angles that each measure $144^\circ$ . The remaining three interior angles have measures in the ratio $3:4:5$ . What is the measure, in degrees, of the largest interior angle of this octagon?

Cevabı ve açıklamayı göster

Cevap: 150

Cevap

The measure of the largest interior angle of the octagon is

150^\circ

.

The total sum of the interior angles of an octagon is

(8-2) \times 180^\circ = 1080^\circ

. The sum of the five given angles is

5 \times 144^\circ = 720^\circ

, leaving a sum of

1080^\circ - 720^\circ = 360^\circ

for the remaining three angles. Since these three angles are in the ratio

3:4:5

, we set

3x + 4x + 5x = 360^\circ

, yielding

12x = 360^\circ

and

x = 30^\circ

. The largest of these three angles is

5 \times 30^\circ = 150^\circ

. Comparing

150^\circ

to the other angles of the octagon (which are

144^\circ

,

90^\circ

, and

120^\circ

), the largest interior angle is

150^\circ

.

Adım Adım Çözüm

1

Calculate the total sum of the interior angles of the octagon.

The total sum is

1080^\circ

.

The sum of the interior angles of any convex polygon with

n

sides is given by

(n - 2) \times 180^\circ

. For an octagon,

n = 8

, so the sum is

(8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ

.

2

Calculate the sum of the five given congruent angles.

The sum of these five angles is

720^\circ

.

Since five angles each measure

144^\circ

, their combined sum is

5 \times 144^\circ = 720^\circ

.

3

Determine the sum of the remaining three interior angles.

The sum of the remaining angles is

360^\circ

.

Subtracting the sum of the five congruent angles from the total sum of the octagon's interior angles yields the sum of the remaining three angles:

1080^\circ - 720^\circ = 360^\circ

.

4

Use the ratio

3:4:5

to find the measures of the remaining three angles.

The measures of the three angles are

90^\circ

,

120^\circ

, and

150^\circ

.

Let the measures of the remaining three angles be

3x

,

4x

, and

5x

. Their sum is

3x + 4x + 5x = 12x = 360^\circ

, which gives

x = 30^\circ

. The largest of these three angles is

5 \times 30^\circ = 150^\circ

.

5

Compare the measures of all interior angles of the octagon to find the largest one.

The largest angle is

150^\circ

.

The octagon's interior angles consist of five angles of

144^\circ

, and three angles of

90^\circ

,

120^\circ

, and

150^\circ

. Comparing these values,

150^\circ

is the largest measure.

Anahtar Kavram

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, and ratios can be used to partition a total sum into specific parts.

Tahmini Süre:2m 0s

Soru 2546Soru →

A segment in the coordinate plane has one endpoint at $A(1, 9)$ and its midpoint at $M(5, 6)$ . Let $B$ represent the other endpoint of the segment. What is the distance between point $B$ and the point $C(3, -5)$ ?

14

2\sqrt{10}

10

5\sqrt{5}

\sqrt{5}

Cevabı ve açıklamayı göster

Cevap: $10$

Cevap

The distance between point

B

and point

C

is

10

.

First, we find the coordinates of point

B(x, y)

using the midpoint formula. Since

M(5, 6)

is the midpoint between

A(1, 9)

and

B(x, y)

, we set up the equations:

5 = \frac{1+x}{2}

which simplifies to

10 = 1+x

and

x = 9

; and

6 = \frac{9+y}{2}

which simplifies to

12 = 9+y

and

y = 3

. Thus, point

B

is

(9, 3)

. Next, we apply the distance formula to find the distance between

B(9, 3)

and

C(3, -5)

:

d = \sqrt{(9-3)^2 + (3-(-5))^2} = \sqrt{6^2 + 8^2} = \sqrt{36+64} = \sqrt{100} = 10

. This is the correct distance.

Adım Adım Çözüm

1

Find the coordinates of endpoint

B(x_B, y_B)

using the midpoint formula.

x_B = 9

and

y_B = 3

, so point

B

is

(9, 3)

.

Since

M(5, 6)

is the midpoint of segment

AB

, we can solve for

B

's coordinates using

5 = \frac{1 + x_B}{2}

and

6 = \frac{9 + y_B}{2}

.

2

Calculate the distance between

B(9, 3)

and

C(3, -5)

using the distance formula.

d = 10

Applying the distance formula:

d = \sqrt{(9-3)^2 + (3-(-5))^2} = \sqrt{6^2 + 8^2} = \sqrt{100} = 10

.

Anahtar Kavram

Using the midpoint formula to find a missing endpoint, and then applying the distance formula between two coordinate points.

Soru 2547Soru →

In $\triangle ABC$ , the measure of $\angle A$ is $40^\circ$ . The measure of $\angle B$ is three times the measure of $\angle C$ . What is the measure, in degrees, of $\angle B$ ?

Cevabı ve açıklamayı göster

Cevap: 105

Cevap

The measure of angle B is 105 degrees.

The sum of the interior angles in any triangle is

180^\circ

. By setting the measure of

\angle C

to

x

and the measure of

\angle B

to

3x

, we can write the equation

40 + 3x + x = 180

. Solving for

x

yields

4x = 140

, which simplifies to

x = 35

. The measure of

\angle B

is

3x

, which is

3 \times 35 = 105^\circ

.

Adım Adım Çözüm

1

Set up the equation using the fact that the sum of angles in a triangle is 180 degrees.

m\angle A + m\angle B + m\angle C = 180^\circ

The angles of any triangle in plane geometry sum to 180 degrees.

2

Represent the unknown angles algebraically.

Let

m\angle C = x

, then

m\angle B = 3x

. Substitute

m\angle A = 40^\circ

.

This allows solving for the unknown angles with a single-variable equation.

3

Solve the equation for

x

.

40 + 4x = 180 \implies 4x = 140 \implies x = 35

To find the measure of angle C.

4

Calculate the measure of angle B.

m\angle B = 3(35) = 105^\circ

Angle B is three times angle C, and we need to find the measure of angle B.

Anahtar Kavram

The sum of the interior angles of a triangle is always 180 degrees.

Soru 2548Soru →

A model glider is launched from a hill. Its height $h(t)$ , in meters above the valley floor $t$ seconds after launch, is modeled by the function $h(t) = -0.5t^2 + 3.5t + 10$ . Which of the following is a possible value of $t$ , in seconds, when the glider is at a height of exactly $12$ meters?

\frac{7 + \sqrt{33}}{2}

\frac{7 + \sqrt{65}}{2}

\frac{-7 + \sqrt{33}}{2}

7 + \sqrt{33}

11

Cevabı ve açıklamayı göster

Cevap: $\frac{7 + \sqrt{33}}{2}$

Cevap

\frac{7 + \sqrt{33}}{2}

The correct answer is obtained by setting the height equation

h(t) = 12

, which simplifies to

-0.5t^2 + 3.5t - 2 = 0

. Multiplying by

-2

gives the standard form

t^2 - 7t + 4 = 0

. Applying the quadratic formula with

a = 1

,

b = -7

, and

c = 4

gives

t = \frac{7 \pm \sqrt{33}}{2}

. Thus, the option representing

\frac{7 + \sqrt{33}}{2}

is the correct choice.

Adım Adım Çözüm

1

Set the height function equal to the target height of 12 meters.

-0.5t^2 + 3.5t + 10 = 12

To find when the glider reaches exactly 12 meters, we set the model function equal to 12.

2

Subtract 12 from both sides to set the quadratic equation to zero.

-0.5t^2 + 3.5t - 2 = 0

A quadratic equation must be in standard form

at^2 + bt + c = 0

before applying the quadratic formula.

3

Multiply the entire equation by

-2

to eliminate decimal coefficients.

t^2 - 7t + 4 = 0

Working with integer coefficients reduces calculation errors when applying the quadratic formula. Here,

a = 1

,

b = -7

, and

c = 4

.

4

Apply the quadratic formula

t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

.

t = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(4)}}{2(1)} = \frac{7 \pm \sqrt{49 - 16}}{2} = \frac{7 \pm \sqrt{33}}{2}

The quadratic formula is used to solve quadratic equations that cannot be easily factored using integers.

Anahtar Kavram

Solving quadratic equations with decimal coefficients by converting to standard integer form and applying the quadratic formula.

Soru 2549Soru →

In right triangle $ABC$ , the measure of $\angle B$ is $90^\circ$ , and the measure of $\angle C$ is $30^\circ$ . Point $D$ lies on segment $BC$ such that the measure of $\angle ADC$ is $135^\circ$ . If the length of segment $AD$ is $8$ units, what is the length, in units, of segment $AC$ ?

4\sqrt{6}

8

8\sqrt{2}

16

16\sqrt{2}

Cevabı ve açıklamayı göster

Cevap: $8\sqrt{2}$

Cevap

8\sqrt{2}

The correct answer is

8\sqrt{2}

. Since points

B

,

D

, and

C

lie on a straight line, the adjacent angles

\angle ADB

and

\angle ADC

must sum to

180^\circ

. Subtracting the given angle measure shows that

\angle ADB = 180^\circ - 135^\circ = 45^\circ

. In the right triangle

ABD

, since

\angle B = 90^\circ

and

\angle ADB = 45^\circ

, the triangle is a

45^\circ-45^\circ-90^\circ

special right triangle. Using the ratio of side lengths for this triangle type, the leg

AB

is equal to the hypotenuse

AD

divided by

\sqrt{2}

, which simplifies to

AB = \frac{8}{\sqrt{2}} = 4\sqrt{2}

. Next, looking at the larger right triangle

ABC

, the angle at

C

is given as

30^\circ

, which makes

\triangle ABC

a

30^\circ-60^\circ-90^\circ

special right triangle. In this type of triangle, the hypotenuse

AC

is twice the length of the shorter leg

AB

, which is opposite the

30^\circ

angle. Multiplying the length of

AB

by

2

yields

AC = 2 \times 4\sqrt{2} = 8\sqrt{2}

units.

Adım Adım Çözüm

1

Find the measure of angle

ADB

using the supplementary angle relationship along the line segment

BC

.

\angle ADB = 180^\circ - 135^\circ = 45^\circ

Points

B

,

D

, and

C

are collinear, meaning

\angle ADB

and

\angle ADC

form a linear pair and must sum to

180^\circ

.

2

Determine the properties of right triangle

ABD

and solve for the length of side

AB

.

\triangle ABD

is a

45^\circ-45^\circ-90^\circ

right triangle, where leg

AB = \frac{AD}{\sqrt{2}} = \frac{8}{\sqrt{2}} = 4\sqrt{2}

units.

Since

\angle B = 90^\circ

and

\angle ADB = 45^\circ

, the remaining angle

\angle DAB

is also

45^\circ

. In a

45^\circ-45^\circ-90^\circ

triangle, the leg length equals the hypotenuse divided by

\sqrt{2}

.

3

Use the properties of the larger

30^\circ-60^\circ-90^\circ

right triangle

ABC

to find the length of hypotenuse

AC

.

AC = 2 \times AB = 2 \times 4\sqrt{2} = 8\sqrt{2}

units.

In right triangle

ABC

, the angle opposite leg

AB

is

\angle C = 30^\circ

. In any

30^\circ-60^\circ-90^\circ

right triangle, the hypotenuse is exactly twice the length of the leg opposite the

30^\circ

angle.

Anahtar Kavram

Using multi-step properties of special right triangles (

45^\circ-45^\circ-90^\circ

and

30^\circ-60^\circ-90^\circ

) sharing a common boundary line.

Soru 2550Soru →

The measures of the five exterior angles of a convex pentagon are in the ratio $2:3:4:4:5$ . What is the measure of the largest interior angle of this pentagon?

80^\circ

100^\circ

120^\circ

140^\circ

150^\circ

Cevabı ve açıklamayı göster

Cevap: $140^\circ$

Cevap

The measure of the largest interior angle of the pentagon is

140^\circ

.

The correct answer is

140^\circ

. The sum of the exterior angles of any convex polygon is

360^\circ

. For a pentagon with exterior angles in the ratio

2:3:4:4:5

, the sum of the ratio parts is

2+3+4+4+5=18

parts. Each part is equal to

360^\circ / 18 = 20^\circ

. The smallest exterior angle has

2

parts, which measures

2 \times 20^\circ = 40^\circ

. Since each interior angle is supplementary to its corresponding exterior angle, the largest interior angle corresponds to the smallest exterior angle. Thus, the largest interior angle measures

180^\circ - 40^\circ = 140^\circ

.

Adım Adım Çözüm

1

Determine the sum of the exterior angles of a convex pentagon.

The sum of the exterior angles of any convex polygon is

360^\circ

.

This is a fundamental property of convex polygons and provides the total value to distribute among the ratio parts.

2

Calculate the measure of the smallest exterior angle using the given ratio of

2:3:4:4:5

.

The sum of the ratio parts is

2+3+4+4+5=18

. The smallest exterior angle has

2

parts, so its measure is

\frac{2}{18} \times 360^\circ = 40^\circ

.

The largest interior angle will be adjacent (and supplementary) to the smallest exterior angle.

3

Calculate the measure of the largest interior angle by subtracting the smallest exterior angle from

180^\circ

.

The largest interior angle is

180^\circ - 40^\circ = 140^\circ

.

An interior angle and its adjacent exterior angle are supplementary and add up to

180^\circ

.

Anahtar Kavram

The sum of the exterior angles of any convex polygon is

360^\circ

. An interior angle and its adjacent exterior angle are supplementary, meaning they add up to

180^\circ

. Consequently, the largest interior angle corresponds to the smallest exterior angle.

Tahmini Süre:1m 30s

Soru 2551Soru →

An irregular convex hexagon has two interior angles that are right angles. The remaining four interior angles have measures in the ratio $4:5:5:6$ . What is the measure, in degrees, of the largest interior angle of this hexagon?

Cevabı ve açıklamayı göster

Cevap: 162

Cevap

The correct answer is

162

degrees. The sum of the interior angles of a hexagon is

720^\circ

. Subtracting the two right angles (

180^\circ

total) leaves a remaining sum of

540^\circ

. The ratio of the remaining four angles is

4:5:5:6

, which can be represented as

4x, 5x, 5x, 6x

, summing to

20x = 540

. Solving for the multiplier gives

x = 27

. The largest angle is

6(27) = 162^\circ

, which is also greater than the two

90^\circ

angles.

The sum of the interior angles of a hexagon is

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

. Subtracting the two right angles (

180^\circ

) gives a remaining sum of

540^\circ

for the other four angles. Let these four angles be

4x, 5x, 5x,

and

6x

. Their sum is

20x = 540

, which solves to

x = 27

. The largest angle is

6x = 6(27) = 162^\circ

, which is also larger than the two

90^\circ

angles.

Adım Adım Çözüm

1

Calculate the sum of all interior angles of a convex hexagon.

The sum is

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

.

The sum of the interior angles of any convex

n

-gon is given by

(n-2) \times 180^\circ

.

2

Subtract the sum of the two right angles from the total sum.

The remaining sum is

720^\circ - 180^\circ = 540^\circ

.

Two right angles contribute

90^\circ + 90^\circ = 180^\circ

to the total.

3

Set up a linear equation representing the ratio of the remaining four angles.

The equation is

4x + 5x + 5x + 6x = 540

, which simplifies to

20x = 540

, yielding

x = 27

.

The angles are proportional to the parts of the ratio, and their sum must equal the remaining

540^\circ

.

4

Calculate the largest angle from the ratio and compare with the right angles.

The largest angle is

6 \times 27 = 162^\circ

.

The largest term in the ratio is

6

, and the resulting angle

162^\circ

is larger than both

90^\circ

and the other calculated angles (

108^\circ

and

135^\circ

).

Anahtar Kavram

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, and ratio relationships can be solved using algebraic multipliers.

Soru 2552Soru →

For all real values of $n$ that satisfy the inequality $\frac{3 - 2n}{4} < \frac{3n - 22}{5}$ , what is the smallest possible integer value of $n$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The smallest integer value of

n

that satisfies the inequality is 5.

Solving the inequality yields

n > \frac{103}{22} \approx 4.68

. The smallest integer greater than

4.68

is

5

.

Adım Adım Çözüm

1

Multiply both sides by 20 to clear the denominators.

5(3 - 2n) < 4(3n - 22)

To eliminate the fractions and simplify the inequality.

2

Distribute the constants on both sides.

15 - 10n < 12n - 88

To remove the parentheses.

3

Subtract

12n

from both sides.

15 - 22n < -88

To group the terms containing the variable on the left side.

4

Subtract 15 from both sides.

-22n < -103

To isolate the variable term on the left side.

5

Divide both sides by

-22

and reverse the inequality sign.

n > \frac{103}{22}

Dividing by a negative number reverses the direction of the inequality sign.

6

Convert the fraction to a decimal to identify the boundary.

n > 4.68

To find the smallest integer value that satisfies this condition.

7

Identify the smallest integer greater than

4.68

.

5

The smallest integer greater than

4.68

is

5

.

Anahtar Kavram

Solving linear inequalities by applying inverse operations and reversing the inequality sign when multiplying or dividing by a negative number.

Tahmini Süre:1m 30s

Soru 2553Soru →

An irregular convex decagon (10-sided polygon) has four interior angles that each measure $150^\circ$ . The remaining six interior angles are congruent to each other. What is the degree measure of each of these remaining six angles?

Cevabı ve açıklamayı göster

Cevap: 140

Cevap

The measure of each of the remaining six interior angles is

140^\circ

.

The total sum of the interior angles of a 10-sided convex polygon is

(10-2) \times 180^\circ = 1,440^\circ

. Subtracting the sum of the four angles that each measure

150^\circ

(

4 \times 150^\circ = 600^\circ

) leaves

840^\circ

for the remaining six angles. Since these remaining six angles are congruent, each measures

840^\circ \div 6 = 140^\circ

.

Adım Adım Çözüm

1

Calculate the total sum of the interior angles of a convex decagon.

1,440^\circ

The interior angle sum of a polygon with

n

sides is

(n-2) \times 180^\circ

, and a decagon has 10 sides.

2

Find the sum of the four angles measuring

150^\circ

.

600^\circ

Multiply the number of angles by their given degree measure.

3

Determine the sum of the remaining six congruent angles.

840^\circ

Subtract the sum of the four known angles from the total interior angle sum of the decagon.

4

Divide the remaining sum by the number of congruent angles.

140^\circ

Since the remaining six angles are equal in measure, dividing their sum by 6 yields the measure of each individual angle.

Anahtar Kavram

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

.

Soru 2554Soru →

In the standard $(x, y)$ coordinate plane, square $PQRS$ has adjacent vertices at $P(1, 2)$ and $Q(4, 6)$ . The line containing the side $QR$ has a y-intercept of $b$ . What is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The correct answer is 9.

The slope of segment

PQ

is

\frac{6 - 2}{4 - 1} = \frac{4}{3}

. Since adjacent sides of a square are perpendicular, the line containing

QR

is perpendicular to

PQ

and passes through

Q(4, 6)

. The slope of this perpendicular line is the negative reciprocal of

\frac{4}{3}

, which is

-\frac{3}{4}

. Using the slope-intercept form

y = mx + b

with point

Q(4, 6)

, we substitute the values:

6 = -\frac{3}{4}(4) + b \Rightarrow 6 = -3 + b \Rightarrow b = 9

.

Adım Adım Çözüm

1

Calculate the slope of side

PQ

using the coordinates of

P(1, 2)

and

Q(4, 6)

.

The slope of

PQ

is

m_{PQ} = \frac{6 - 2}{4 - 1} = \frac{4}{3}

.

To determine the direction of side

PQ

so we can find the perpendicular slope for

QR

.

2

Find the slope of the line containing side

QR

.

The slope of

QR

is

m_{QR} = -\frac{3}{4}

.

Because adjacent sides of a square are perpendicular, the slope of

QR

is the negative reciprocal of the slope of

PQ

.

3

Find the equation of the line containing

QR

using the slope-intercept form and the coordinates of vertex

Q(4, 6)

.

Substituting the slope

m = -\frac{3}{4}

and point

(4, 6)

into

y = mx + b

gives

6 = -\frac{3}{4}(4) + b

, which simplifies to

6 = -3 + b

, so

b = 9

.

The line containing side

QR

must pass through vertex

Q

, which allows us to determine the y-intercept

b

.

Anahtar Kavram

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

m_1 \cdot m_2 = -1

.

Soru 2555Soru →

In the standard $(x, y)$ coordinate plane, a trapezoid has vertices at $A(-5, -4)$ , $B(7, -4)$ , $C(7, 1)$ , and $D(-5, 6)$ . What is the perimeter of this trapezoid?

Cevabı ve açıklamayı göster

Cevap: 40

Cevap

The perimeter of the trapezoid is 40.

The perimeter of the trapezoid is found by summing the lengths of its four sides. Two sides are vertical (AD of length 10 and BC of length 5), one side is horizontal (AB of length 12), and the final side is skewed (CD of length 13, found using the distance formula). Summing these gives 10 + 5 + 12 + 13 = 40.

Adım Adım Çözüm

1

Calculate the lengths of the vertical sides AD and BC

AD = 10, BC = 5

Since vertices A(-5, -4) and D(-5, 6) share the same x-coordinate of -5, the side AD is vertical. Its length is the difference in their y-coordinates: 6 - (-4) = 10. Similarly, vertices B(7, -4) and C(7, 1) share the same x-coordinate of 7, so the side BC is vertical. Its length is the difference in their y-coordinates: 1 - (-4) = 5.

2

Calculate the length of the horizontal side AB

AB = 12

Since vertices A(-5, -4) and B(7, -4) share the same y-coordinate of -4, the side AB is horizontal. Its length is the difference in their x-coordinates: 7 - (-5) = 12.

3

Calculate the length of the skewed side CD using the distance formula

CD = 13

Using the coordinates of C(7, 1) and D(-5, 6), the distance is calculated as: CD = \sqrt{(-5 - 7)^2 + (6 - 1)^2} = \sqrt{(-12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13.

4

Sum the lengths of all sides to find the perimeter

Perimeter = 40

The perimeter of a polygon is the sum of the lengths of all its sides: Perimeter = AB + BC + CD + DA = 12 + 5 + 13 + 10 = 40.

Anahtar Kavram

Calculating side lengths of a geometric figure on a coordinate plane to find its perimeter

Soru 2556Soru →

Which of the following inequalities represents the complete solution set for $x$ in the inequality $\frac{x}{3} - \frac{2x - 5}{2} \ge 2$ ?

x \le \frac{3}{4}

x \ge \frac{3}{4}

x \le 3

x \le -\frac{27}{4}

x \ge 3

Cevabı ve açıklamayı göster

Cevap: $x \le \frac{3}{4}$

Cevap

The inequality is satisfied by all values of

x

less than or equal to

\frac{3}{4}

.

The correct solution is obtained by rewriting the fractions with a common denominator of 6, which yields

\frac{2x - 3(2x - 5)}{6} \ge 2

. Distributing the negative sign gives

\frac{-4x + 15}{6} \ge 2

. Multiplying both sides by 6 and subtracting 15 results in

-4x \ge -3

. Dividing by

-4

and reversing the inequality sign yields

x \le \frac{3}{4}

.

Adım Adım Çözüm

1

Find a common denominator of 6 for the two fractions on the left side of the inequality.

\frac{2x}{6} - \frac{3(2x - 5)}{6} \ge 2

Finding a common denominator is necessary to combine fractional terms with different denominators.

2

Combine the fractions into a single expression, distributing the subtraction sign to both terms in the numerator of the second fraction.

\frac{2x - (6x - 15)}{6} \ge 2 \implies \frac{-4x + 15}{6} \ge 2

Combining the fractions simplifies the inequality. Distributing the negative sign ensures correct operations on the terms.

3

Multiply both sides of the inequality by 6.

-4x + 15 \ge 12

Multiplying by a positive number isolates the numerator without changing the direction of the inequality sign.

4

Subtract 15 from both sides of the inequality.

-4x \ge -3

This isolates the variable term on the left side.

5

Divide both sides by

-4

and reverse the direction of the inequality sign.

x \le \frac{3}{4}

Dividing both sides of an inequality by a negative number requires reversing the inequality sign to maintain the correct relationship.

Anahtar Kavram

Solving linear inequalities requires applying standard algebraic operations while remembering to reverse the inequality sign when multiplying or dividing both sides by a negative number.

Alternatif Yöntem

Instead of combining fractions first, multiply every term on both sides of the inequality by 6 (the least common multiple of the denominators) to eliminate the fractions immediately:

6 \cdot \left(\frac{x}{3}\right) - 6 \cdot \left(\frac{2x - 5}{2}\right) \ge 6 \cdot 2

. This simplifies directly to

2x - 3(2x - 5) \ge 12

, which reduces to

-4x + 15 \ge 12

.

Tahmini Süre:1m 30s

Soru 2557Soru →

On a coordinate map of an airport, Runway A is represented by the line $4x - 3y = 12$ . Runway B is designed to be perpendicular to Runway A, and its path crosses the $y$ -axis at $(0, 7)$ . If Runway B passes through a guidance beacon located at the coordinate $(p, 1)$ , what is the value of $p$ ?

-8

-\frac{9}{2}

-\frac{32}{3}

8

\frac{9}{2}

Cevabı ve açıklamayı göster

Cevap: $8$

Cevap

8

The correct answer is

8

. First, find the slope of Runway A by rewriting

4x - 3y = 12

as

y = \frac{4}{3}x - 4

, which gives a slope of

\frac{4}{3}

. The slope of Runway B must be the negative reciprocal,

-\frac{3}{4}

. Using the given

y

-intercept of

7

, the equation for Runway B is

y = -\frac{3}{4}x + 7

. Substituting the point

(p, 1)

gives

1 = -\frac{3}{4}p + 7

. Subtracting

7

from both sides yields

-6 = -\frac{3}{4}p

. Multiplying both sides by

-\frac{4}{3}

gives

p = 8

.

Adım Adım Çözüm

1

Convert the equation of Runway A,

4x - 3y = 12

, into slope-intercept form (

y = mx + b

) to find its slope.

The equation becomes

y = \frac{4}{3}x - 4

, which shows the slope of Runway A is

m_1 = \frac{4}{3}

.

Finding the slope of the first line is necessary to determine the slope of any perpendicular line.

2

Calculate the perpendicular slope (

m_2

) for Runway B by taking the negative reciprocal of the slope of Runway A (

m_1

).

The perpendicular slope is

m_2 = -\frac{1}{m_1} = -\frac{3}{4}

.

Perpendicular lines in a coordinate plane have slopes that are negative reciprocals of each other.

3

Write the equation of Runway B using its slope

m_2 = -\frac{3}{4}

and its

y

-intercept of

7

.

The equation of Runway B is

y = -\frac{3}{4}x + 7

.

The slope-intercept form of a line is

y = mx + b

, where

m

is the slope and

b

is the

y

-intercept.

4

Substitute the point

(p, 1)

into the equation of Runway B and solve for

p

.

1 = -\frac{3}{4}p + 7 \implies -6 = -\frac{3}{4}p \implies 24 = 3p \implies p = 8

.

Since the guidance beacon lies on Runway B, its coordinates must satisfy the equation of the line.

Anahtar Kavram

Perpendicular lines have slopes that are negative reciprocals of each other, meaning their product is

-1

. Once the perpendicular slope and y-intercept are known, the line's equation can be written and solved for a missing coordinate.

Tahmini Süre:1m 0s

Soru 2558Soru →

An irregular convex hexagon has three interior angles that each measure $110^\circ$ . The remaining three interior angles have measures in the ratio $3:4:5$ . What is the degree measure of the exterior angle corresponding to the smallest interior angle of this hexagon?

17.5^\circ

27.5^\circ

50.0^\circ

82.5^\circ

97.5^\circ

Cevabı ve açıklamayı göster

Cevap: $82.5^\circ$

Cevap

82.5^\circ

The sum of the interior angles of a hexagon is

720^\circ

. Subtracting the three known angles (

330^\circ

) leaves

390^\circ

for the remaining three angles. Using the ratio

3:4:5

, the smallest of these remaining angles is calculated as

\frac{3}{12} \times 390^\circ = 97.5^\circ

. Since

97.5^\circ

is smaller than the other angles (

110^\circ

,

130^\circ

,

162.5^\circ

), it is the smallest interior angle of the hexagon. The exterior angle is supplementary to the interior angle, so it measures

180^\circ - 97.5^\circ = 82.5^\circ

.

Adım Adım Çözüm

1

Calculate the sum of all interior angles of the hexagon.

The sum is

(6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ

.

This establishes the total measure of the interior angles of a hexagon.

2

Subtract the sum of the three known interior angles from the total sum.

The remaining sum is

720^\circ - (3 \times 110^\circ) = 720^\circ - 330^\circ = 390^\circ

.

This determines the combined measure of the three remaining angles.

3

Determine the measures of the remaining three angles using the ratio.

Setting up the equation

3y + 4y + 5y = 390^\circ

gives

12y = 390^\circ

, or

y = 32.5^\circ

. The individual angles are

3 \times 32.5^\circ = 97.5^\circ

,

4 \times 32.5^\circ = 130^\circ

, and

5 \times 32.5^\circ = 162.5^\circ

.

This finds each of the remaining interior angle measures.

4

Find the smallest interior angle and compute its supplementary exterior angle.

The smallest interior angle of the hexagon is

97.5^\circ

. The corresponding exterior angle is

180^\circ - 97.5^\circ = 82.5^\circ

.

An interior angle and its adjacent exterior angle sum to

180^\circ

.

Anahtar Kavram

Polygon Angles and Properties

Soru 2559Soru →

For what value of $c$ does the quadratic equation $0.5x^2 - 3x + c = 0$ have two real solutions that differ by exactly 4?

-2.5

2.5

4

5

10

Cevabı ve açıklamayı göster

Cevap: 2.5

Cevap

2.5

The correct value is 2.5. By utilizing the formula for the difference of the roots,

\frac{\sqrt{b^2 - 4ac}}{|a|} = 4

, and substituting

a = 0.5

and

b = -3

, we get

\frac{\sqrt{9 - 2c}}{0.5} = 4

. This simplifies to

\sqrt{9 - 2c} = 2

, which squares to

9 - 2c = 4

. Solving for

c

yields 2.5.

Adım Adım Çözüm

1

Identify the coefficients and apply the relationship for the difference between two roots.

For a quadratic equation

ax^2 + bx + c = 0

, the roots

x_1

and

x_2

satisfy

|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

. Here,

a = 0.5

,

b = -3

, and the difference is 4.

This formula relates the difference of the roots directly to the coefficients of the quadratic equation.

2

Substitute the given values into the formula and solve for

c

.

Substituting the values gives

\frac{\sqrt{(-3)^2 - 4(0.5)c}}{|0.5|} = 4 \Rightarrow \frac{\sqrt{9 - 2c}}{0.5} = 4

. Multiplying both sides by 0.5 yields

\sqrt{9 - 2c} = 2

. Squaring both sides gives

9 - 2c = 4

.

Simplifying the equation isolates the variable

c

under the radical.

3

Complete the algebraic isolation to find the final value of

c

.

2c = 5 \Rightarrow c = 2.5

.

This final step solves the linear equation for

c

.

Anahtar Kavram

Using the discriminant and properties of roots to solve quadratic equations with given constraints.

Soru 2560Soru →

In the standard $(x, y)$ coordinate plane, the line $L_1$ is parallel to the line $y = 3x - 7$ . The line $L_2$ is perpendicular to $L_1$ and passes through the point $(6, 5)$ . If the $y$ -intercept of $L_2$ is $(0, c)$ , what is the value of $c$ ?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of

c

is

7

.

Since the line

L_1

is parallel to

y = 3x - 7

, its slope is

3

. The line

L_2

is perpendicular to

L_1

, so its slope is the negative reciprocal of

3

, which is

-\frac{1}{3}

. Using the point-slope formula with the point

(6, 5)

yields the equation of

L_2

:

y - 5 = -\frac{1}{3}(x - 6)

, which simplifies to

y = -\frac{1}{3}x + 7

. The

y

-intercept of this line is

(0, 7)

, which gives

c = 7

.

Adım Adım Çözüm

1

Determine the slope of line

L_1

.

Slope of

L_1

is

3

.

Parallel lines have equal slopes, and the given reference line

y = 3x - 7

has a slope of

3

.

2

Determine the slope of line

L_2

.

Slope of

L_2

is

-\frac{1}{3}

.

Perpendicular lines have slopes that are negative reciprocals of each other.

3

Find the equation of line

L_2

.

The equation is

y = -\frac{1}{3}x + 7

.

Use the point-slope form

y - y_1 = m(x - x_1)

with point

(6, 5)

and slope

-\frac{1}{3}

.

4

Identify the

y

-intercept constant

c

.

c = 7

.

The equation of

L_2

is in slope-intercept form

y = mx + b

, meaning the

y

-intercept is

(0, 7)

.

Anahtar Kavram

The relationship between the slopes of parallel lines (which are equal) and perpendicular lines (which are negative reciprocals).

Cevap

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Sık Yapılan Hatalar

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Alternatif Yöntem

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Alternatif Yöntem

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Alternatif Yöntem

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