Tüm alıştırma soruları

Soru 401Soru →

In rhombus $ABCD$ , the perimeter is $100$ and the length of diagonal $BD$ is $30$ . Point $P$ lies on diagonal $AC$ such that the ratio of the length of segment $AP$ to the length of segment $PC$ is $3:7$ . What is the length of segment $BP$ ?

Cevabı ve açıklamayı göster

Cevap: 17

Cevap

The length of segment

BP

is 17.

The correct answer is found by utilizing the properties of a rhombus. A rhombus has four congruent sides, meaning each side of a rhombus with perimeter

100

has a length of

25

. The diagonals of a rhombus are perpendicular bisectors of one another. Letting

O

be the intersection of the diagonals, we find

BO = 15

since diagonal

BD = 30

. Using the Pythagorean theorem on right triangle

AOB

, we determine that the other half-diagonal is

AO = \sqrt{25^2 - 15^2} = 20

, which means the full diagonal

AC = 40

. Point

P

divides

AC

in the ratio

3:7

, meaning

AP = 12

and

PC = 28

. The distance from

P

to the intersection point

O

is

OP = AO - AP = 20 - 12 = 8

. Finally, applying the Pythagorean theorem to right triangle

BOP

with legs

BO = 15

and

OP = 8

yields

BP = \sqrt{15^2 + 8^2} = 17

.

Adım Adım Çözüm

1

Calculate the side length of rhombus

ABCD

from its perimeter.

Each side length is

25

.

A rhombus has four equal sides, so the side length is the perimeter divided by four:

\frac{100}{4} = 25

.

2

Find the length of half of diagonal

BD

.

BO = 15

, where

O

is the intersection of diagonals

AC

and

BD

.

The diagonals of a rhombus bisect each other.

3

Calculate the half-diagonal length

AO

and full diagonal length

AC

.

AO = 20

and

AC = 40

.

The diagonals of a rhombus are perpendicular, forming right triangle

AOB

. By the Pythagorean theorem,

AO = \sqrt{AB^2 - BO^2} = \sqrt{25^2 - 15^2} = 20

. Since the diagonals bisect each other, the total length of diagonal

AC

is

2 \times 20 = 40

.

4

Determine the length of segment

AP

.

AP = 12

.

Point

P

lies on diagonal

AC

such that the ratio of segment

AP

to

PC

is

3:7

. Therefore,

AP = \frac{3}{3+7} \times AC = \frac{3}{10} \times 40 = 12

.

5

Find the distance

OP

between point

P

and the intersection point

O

.

OP = 8

.

Since

AO = 20

and

P

is

12

units from

A

,

P

lies on the segment

AO

. Thus, the distance from

P

to

O

is

OP = AO - AP = 20 - 12 = 8

.

6

Calculate the length of segment

BP

.

BP = 17

.

Because the diagonals of a rhombus are perpendicular,

\triangle BOP

is a right triangle with legs

BO = 15

and

OP = 8

. Using the Pythagorean theorem,

BP = \sqrt{BO^2 + OP^2} = \sqrt{15^2 + 8^2} = 17

.

Anahtar Kavram

Properties of Rhombuses (perpendicular bisecting diagonals, equal side lengths) and the Pythagorean Theorem

Tahmini Süre:2m 0s

Soru 402Soru →

In the standard $(x, y)$ coordinate plane, three vertices of a rectangle are $A(-4, 1)$ , $B(2, 9)$ , and $C(6, 6)$ . What is the area of the rectangle, in square units?

Cevabı ve açıklamayı göster

Cevap: 50

Cevap

50

The length of side

AB

is calculated as

\sqrt{(2 - (-4))^2 + (9 - 1)^2} = \sqrt{36 + 64} = 10

. The length of the adjacent side

BC

is calculated as

\sqrt{(6 - 2)^2 + (6 - 9)^2} = \sqrt{16 + 9} = 5

. The area of the rectangle is the product of these two perpendicular side lengths, which is

10 \times 5 = 50

.

Adım Adım Çözüm

1

Calculate the length of side

AB

using the distance formula.

AB = 10

To find one of the side lengths of the rectangle.

2

Calculate the length of side

BC

using the distance formula.

BC = 5

To find the adjacent side length of the rectangle.

3

Multiply the two adjacent side lengths to find the area of the rectangle.

50

The area of a rectangle is equal to the product of its length and width.

Anahtar Kavram

Calculating the area of a geometric figure on the coordinate plane by determining its side lengths using the distance formula.

Soru 403Soru →

For the quadratic equation $0.4x^2 + bx - 4.8 = 0$ , where $b$ is a constant, the sum of the two solutions is equal to the product of the two solutions. What is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: 4.8

Cevap

The value of the constant

b

is

4.8

.

According to Vieta's formulas, the sum of the solutions to the quadratic equation

ax^2 + bx + c = 0

is given by

-\frac{b}{a}

and their product is given by

\frac{c}{a}

. For the given equation

0.4x^2 + bx - 4.8 = 0

, the product of the solutions is

\frac{-4.8}{0.4} = -12

. Setting the sum equal to the product yields the equation

-\frac{b}{0.4} = -12

. Multiplying both sides by

-0.4

isolates

b

, giving

b = 4.8

.

Adım Adım Çözüm

1

Identify the coefficients of the quadratic equation.

a = 0.4

,

b = b

, and

c = -4.8

.

To apply formulas relating the coefficients to the solutions.

2

Express the sum and product of the solutions using Vieta's formulas.

Sum of solutions is

-\frac{b}{0.4}

and product of solutions is

\frac{-4.8}{0.4} = -12

.

To establish the mathematical relationship given in the problem.

3

Equate the sum and product of the solutions and solve for the constant

b

.

-\frac{b}{0.4} = -12 \implies b = -12 \times (-0.4) = 4.8

.

The problem states that the sum of the two solutions is equal to their product.

Anahtar Kavram

Sum and Product of Roots (Vieta's Formulas)

Soru 404Soru →

In the standard $(x, y)$ coordinate plane, the vertices of a right triangle are $P(1, 2)$ , $Q(5, 5)$ , and $R(k, 9)$ . If the right angle of the triangle is at vertex $Q$ , what is the value of the constant $k$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of

k

is

2

.

Because the right angle of the triangle is at vertex

Q

, segment

PQ

must be perpendicular to segment

QR

. The slope of

PQ

is

\frac{5 - 2}{5 - 1} = \frac{3}{4}

. The slope of a perpendicular line is the negative reciprocal, so the slope of

QR

must be

-\frac{4}{3}

. Expressing the slope of

QR

using the coordinates of

Q(5, 5)

and

R(k, 9)

gives

\frac{9 - 5}{k - 5} = \frac{4}{k - 5}

. Setting this equal to

-\frac{4}{3}

and solving for

k

yields

k - 5 = -3

, which means

k = 2

.

Adım Adım Çözüm

1

Use the slope formula

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

to calculate the slope of the line segment

PQ

with endpoints

P(1, 2)

and

Q(5, 5)

.

m_{PQ} = \frac{5 - 2}{5 - 1} = \frac{3}{4}

This establishes the direction of the first leg of the right triangle.

2

Find the slope of segment

QR

. Because the right angle is at vertex

Q

, the segment

PQ

is perpendicular to segment

QR

.

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

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

m_1 \cdot m_2 = -1

).

3

Write the slope of segment

QR

in terms of

k

using coordinates

Q(5, 5)

and

R(k, 9)

.

m_{QR} = \frac{9 - 5}{k - 5} = \frac{4}{k - 5}

This sets up an equation to find the unknown coordinate value.

4

Equate the two expressions for the slope of

QR

and solve for

k

.

\frac{4}{k - 5} = -\frac{4}{3} \Rightarrow k - 5 = -3 \Rightarrow k = 2

Solving the rational equation yields the correct coordinate parameter.

Anahtar Kavram

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

Tahmini Süre:1m 30s

Soru 405Soru →

An angle $\theta$ in standard position is rotated counterclockwise by $225^\circ$ . The terminal ray of the resulting angle lies in the fourth quadrant along the line $y = -x\sqrt{3}$ . If the radian measure of the smallest positive angle $\theta$ is written in simplest form as $\frac{a\pi}{b}$ , where $a$ and $b$ are positive integers, what is the value of $a + b$ ?

Cevabı ve açıklamayı göster

Cevap: 17

Cevap

The value of

a + b

is

17

.

The resulting angle

\alpha

lies along the line

y = -x\sqrt{3}

in the fourth quadrant, meaning its measure is

300^\circ

plus any multiple of

360^\circ

. Subtracting the counterclockwise rotation of

225^\circ

gives the original angle

\theta = 75^\circ

(for the smallest positive angle). Converting

75^\circ

to radians by multiplying by

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

yields

\frac{5\pi}{12}

. Since the fraction is in simplest form,

a = 5

and

b = 12

, and their sum is

17

.

Adım Adım Çözüm

1

Find the angle of the terminal ray after rotation from its equation and quadrant

The terminal ray after rotation is at an angle of

300^\circ

(or

\frac{5\pi}{3}

radians)

The line

y = -x\sqrt{3}

has a slope of

-\sqrt{3}

, so the angle

\alpha

in Quadrant IV satisfies

\tan\alpha = -\sqrt{3}

, which means

\alpha = 300^\circ

.

2

Set up and solve the equation for the original angle

\theta

before the counterclockwise rotation of

225^\circ

\theta = 75^\circ + 360^\circ k

Since the angle was rotated counterclockwise by

225^\circ

to reach the final position of

300^\circ

, we have

\theta + 225^\circ = 300^\circ + 360^\circ k

.

3

Determine the smallest positive value of

\theta

\theta = 75^\circ

Setting

k = 0

gives the smallest positive angle of

75^\circ

.

4

Convert the angle

\theta

from degrees to radians

\theta = \frac{5\pi}{12}

radians

To convert degrees to radians, multiply by

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

, yielding

\frac{75\pi}{180} = \frac{5\pi}{12}

.

5

Calculate the sum of the numerator and denominator of the simplified radian fraction

a + b = 17

The fraction

\frac{5}{12}

is in simplest form, so

a = 5

and

b = 12

. The sum is

5 + 12 = 17

.

Anahtar Kavram

Converting degree measures to radian measures and finding coterminal angles on the unit circle

Soru 406Soru →

A convex octagon has interior angles whose measures, in degrees, are eight consecutive even integers. What is the measure, in degrees, of the largest interior angle of this octagon?

Cevabı ve açıklamayı göster

Cevap: 142

Cevap

The correct answer is 142.

The sum of the interior angles of a convex octagon is

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

. If we represent the eight consecutive even integer angle measures as

x, x+2, x+4, x+6, x+8, x+10, x+12,

and

x+14

, their sum is

8x + 56

. Setting this equal to

1080^\circ

and solving for

x

yields

x = 128

. The largest angle is

x + 14

, which equals

128 + 14 = 142^\circ

.

Adım Adım Çözüm

1

Calculate the sum of the interior angles of a convex octagon.

The sum of the interior angles is

1080^\circ

.

The formula for the sum of the interior angles of an

n

-sided polygon is

(n-2) \times 180^\circ

. For an octagon (

n=8

), the sum is

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

.

2

Set up an equation representing the sum of the eight consecutive even integer angle measures.

The equation is

8x + 56 = 1080

.

Letting the smallest angle measure be

x

, the eight consecutive even integer angle measures are

x, x+2, x+4, x+6, x+8, x+10, x+12,

and

x+14

. Their sum is

8x + 56

, which must equal the total sum of the interior angles (

1080^\circ

).

3

Solve the equation for the smallest angle measure,

x

.

x = 128

Subtracting 56 from both sides of the equation yields

8x = 1024

. Dividing both sides by 8 gives

x = 128

.

4

Calculate the measure of the largest interior angle.

The measure of the largest angle is

142^\circ

.

The largest angle is represented by the expression

x + 14

. Substituting

128

for

x

gives

128 + 14 = 142

.

Anahtar Kavram

Calculating the sum of the interior angles of a convex polygon and using algebraic methods to find unknown angle measures.

Soru 407Soru →

In the standard $(x, y)$ coordinate plane, a line passes through the points $(a, 2)$ and $(10, a - 1)$ . If the slope of the line is $-\frac{1}{3}$ , what is the value of $a$ ?

Cevabı ve açıklamayı göster

Cevap: -0.5

Cevap

The value of

a

is

-0.5

(or

-\frac{1}{2}

)

Applying the slope formula

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

to the points

(a, 2)

and

(10, a - 1)

with slope

-\frac{1}{3}

yields the equation

-\frac{1}{3} = \frac{a - 3}{10 - a}

. Solving this linear equation correctly yields

a = -0.5

.

Adım Adım Çözüm

1

Apply the slope formula using the given coordinates.

-\frac{1}{3} = \frac{(a - 1) - 2}{10 - a}

The slope 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 numerator.

-\frac{1}{3} = \frac{a - 3}{10 - a}

Subtracting

2

from

a - 1

simplifies the numerator to

a - 3

.

3

Cross-multiply to solve the rational equation.

-1(10 - a) = 3(a - 3)

Multiplying both sides by the denominators eliminates the fractions.

4

Solve the linear equation for

a

.

a = -0.5

Distributing on both sides gives

-10 + a = 3a - 9

. Rearranging terms yields

2a = -1

, which simplifies to

a = -0.5

.

Anahtar Kavram

Slope of a Line

Soru 408Soru →

For an angle $\theta$ in the interval $\frac{3\pi}{2} < \theta < 2\pi$ , the expression $\sec\theta - \tan\theta$ is equal to $3$ . What is the value of $\csc\theta + \cot\theta$ ?

Cevabı ve açıklamayı göster

Cevap: -2

Cevap

The value of

\csc\theta + \cot\theta

is

-2

.

By using the difference of squares on the identity

\sec^2\theta - \tan^2\theta = 1

, we get

(\sec\theta - \tan\theta)(\sec\theta + \tan\theta) = 1

. Substituting

\sec\theta - \tan\theta = 3

gives

\sec\theta + \tan\theta = \frac{1}{3}

. Solving the system of equations gives

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

and

\tan\theta = -\frac{4}{3}

. Since

\theta

lies in Quadrant IV,

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

and

\sin\theta = -\frac{4}{5}

. We then find

\csc\theta = -\frac{5}{4}

and

\cot\theta = -\frac{3}{4}

, which sum to

-2

.

Adım Adım Çözüm

1

Use the Pythagorean identity

\sec^2\theta - \tan^2\theta = 1

, which factors into

(\sec\theta - \tan\theta)(\sec\theta + \tan\theta) = 1

.

Since

\sec\theta - \tan\theta = 3

, we have

3(\sec\theta + \tan\theta) = 1 \implies \sec\theta + \tan\theta = \frac{1}{3}

.

To establish a second linear equation in terms of

\sec\theta

and

\tan\theta

.

2

Add and subtract the two equations:

\sec\theta - \tan\theta = 3

and

\sec\theta + \tan\theta = \frac{1}{3}

.

Adding them gives

2\sec\theta = \frac{10}{3} \implies \sec\theta = \frac{5}{3}

. Subtracting the first from the second gives

2\tan\theta = -\frac{8}{3} \implies \tan\theta = -\frac{4}{3}

.

To isolate the values of

\sec\theta

and

\tan\theta

.

3

Find

\cos\theta

and

\sin\theta

using

\cos\theta = \frac{1}{\sec\theta}

and

\sin\theta = \tan\theta\cos\theta

.

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

and

\sin\theta = -\frac{4}{5}

. Since

\frac{3\pi}{2} < \theta < 2\pi

(Quadrant IV), cosine is positive and sine is negative, which matches these values.

To find the primary trigonometric values needed for the reciprocal functions.

4

Calculate

\csc\theta

and

\cot\theta

using reciprocal identities.

\csc\theta = \frac{1}{\sin\theta} = -\frac{5}{4}

and

\cot\theta = \frac{1}{\tan\theta} = -\frac{3}{4}

.

To obtain the terms of the required sum.

5

Sum the values of

\csc\theta

and

\cot\theta

.

\csc\theta + \cot\theta = -\frac{5}{4} + \left(-\frac{3}{4}\right) = -\frac{8}{4} = -2

.

To obtain the final value requested by the question.

Anahtar Kavram

Pythagorean and Reciprocal Trigonometric Identities

Soru 409Soru →

Three vertices of a parallelogram are $A(1, 3)$ , $B(-2, -1)$ , and $C(4, -1)$ in the standard $(x, y)$ coordinate plane. If the fourth vertex, $D$ , is located in the fourth quadrant, what is the $y$ -coordinate of $D$ ?

Cevabı ve açıklamayı göster

Cevap: -5

Cevap

The

y

-coordinate of the fourth vertex of the parallelogram in the fourth quadrant is

-5

.

By applying the midpoint formula to the diagonals of the three possible parallelograms formed by the vertices

A(1, 3)

,

B(-2, -1)

, and

C(4, -1)

, we find the candidate points for the fourth vertex

D

to be

(7, 3)

,

(-5, 3)

, and

(1, -5)

. The point

(1, -5)

is the only candidate that lies in the fourth quadrant, where

x > 0

and

y < 0

. Therefore, the

y

-coordinate of the fourth vertex is

-5

.

Adım Adım Çözüm

1

Set up equations based on the property that the diagonals of a parallelogram bisect each other (have the same midpoint).

Three possible configurations of diagonals lead to three potential sets of coordinates for the fourth vertex

D(x, y)

:

(7, 3)

,

(-5, 3)

, and

(1, -5)

.

Three points in a coordinate plane can form three distinct parallelograms depending on which pairs are connected as diagonals.

2

Identify the signs of the coordinates for each potential vertex to determine which quadrant it lies in.

The point

(7, 3)

is in Quadrant I (

x > 0, y > 0

). The point

(-5, 3)

is in Quadrant II (

x < 0, y > 0

). The point

(1, -5)

is in Quadrant IV (

x > 0, y < 0

).

The fourth quadrant is defined by positive

x

-values and negative

y

-values.

3

Select the correct coordinate value corresponding to the question's requirement.

The

y

-coordinate of the vertex in the fourth quadrant,

(1, -5)

, is

-5

.

The question asks specifically for the

y

-coordinate of the fourth vertex

D

.

Anahtar Kavram

Finding a missing vertex of a parallelogram on the coordinate plane using midpoint properties

Soru 410Soru →

In the standard $(x, y)$ coordinate plane, line $L_1$ passes through the points $(1, 3)$ and $(4, 8)$ . Line $L_2$ is perpendicular to line $L_1$ . If line $L_2$ passes through the points $(5, k)$ and $(10, 1)$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

k

is

4

.

The correct answer is

4

. First, determine the slope of line

L_1

using the points

(1, 3)

and

(4, 8)

, which is

\frac{8 - 3}{4 - 1} = \frac{5}{3}

. Because line

L_2

is perpendicular to

L_1

, its slope must be the negative reciprocal of

\frac{5}{3}

, which is

-\frac{3}{5}

. Next, set up the slope equation for

L_2

with the points

(5, k)

and

(10, 1)

, yielding

\frac{1 - k}{10 - 5} = -\frac{3}{5}

. Simplifying the equation gives

\frac{1 - k}{5} = -\frac{3}{5}

, which reduces to

1 - k = -3

. Solving for

k

gives

k = 4

.

Adım Adım Çözüm

1

Calculate the slope of line

L_1

using the coordinates of the two given points,

(1, 3)

and

(4, 8)

.

The slope of line

L_1

is

\frac{5}{3}

.

Using the slope formula

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

, we find

m_1 = \frac{8 - 3}{4 - 1} = \frac{5}{3}

.

2

Determine the slope of line

L_2

based on the perpendicular relationship between

L_1

and

L_2

.

The slope of line

L_2

is

-\frac{3}{5}

.

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

\frac{5}{3}

is

-\frac{3}{5}

.

3

Use the coordinates

(5, k)

and

(10, 1)

on line

L_2

to write an expression for its slope, set it equal to

-\frac{3}{5}

, and solve for

k

.

k = 4

The slope expression is

\frac{1 - k}{10 - 5} = \frac{1 - k}{5}

. Setting this equal to the perpendicular slope gives

\frac{1 - k}{5} = -\frac{3}{5}

. Multiplying both sides by

5

results in

1 - k = -3

. Adding

k

to both sides and adding

3

to both sides yields

k = 4

.

Anahtar Kavram

The slopes of perpendicular lines in a coordinate plane are negative reciprocals of each other, meaning their product is

-1

(

m_1 \cdot m_2 = -1

).

Tahmini Süre:1m 15s

Soru 411Soru →

An architect is designing a triangular roof truss, $\triangle ABC$ , where side $AB$ is equal in length to side $AC$ . The height of the truss, represented by the altitude from vertex $A$ to the base $BC$ , is $12$ feet. If the measure of the base angle $\angle ABC$ is $30^\circ$ , what is the length, in feet, of the base $BC$ ? (Round your answer to the nearest tenth.)

Cevabı ve açıklamayı göster

Cevap: 41.6

Cevap

The correct answer is 41.6

The correct answer is obtained by recognizing that the altitude of the isosceles triangle bisects the base into two congruent

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

right triangles. The leg opposite the

30^\circ

angle is

12

feet, so the leg adjacent (which is half the base) is

12\sqrt{3}

feet. Doubling this gives a total base length of

24\sqrt{3}

feet, which is approximately

41.6

feet when rounded to the nearest tenth.

Adım Adım Çözüm

1

Identify the right triangle formed by the altitude.

An altitude

AD

perpendicular to base

BC

, creating two right triangles,

\triangle ABD

and

\triangle ACD

, with

AD = 12

feet.

In an isosceles triangle, the altitude to the base bisects the base and is perpendicular to it.

2

Determine the angles of the right triangle

\triangle ABD

.

Triangle

\triangle ABD

is a

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

special right triangle.

Angle

B

is

30^\circ

and angle

ADB

is

90^\circ

, leaving

60^\circ

for angle

BAD

.

3

Calculate the length of the segment

BD

.

BD = 12\sqrt{3}

feet

In a

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

triangle, the longer leg is

\sqrt{3}

times the shorter leg (which is opposite the

30^\circ

angle).

4

Find the total length of the base

BC

.

BC = 24\sqrt{3}

feet

Since

D

is the midpoint of

BC

, the total length

BC

is

2 \times BD

.

5

Convert the exact value to a decimal rounded to the nearest tenth.

BC \approx 41.6

24 \times 1.73205 = 41.569

, which rounds to

41.6

.

Anahtar Kavram

Properties of special 30-60-90 right triangles and altitudes of isosceles triangles

Soru 412Soru →

An angle $\theta$ lies in the third quadrant, where $\pi < \theta < \frac{3\pi}{2}$ . If $\sin\theta - \cos\theta = -\frac{1}{5}$ , what is the value of the expression $125(\sin^3\theta + \cos^3\theta)$ ?

Cevabı ve açıklamayı göster

Cevap: -91

Cevap

The correct value of the expression is -91.

Squaring the equation

\sin\theta - \cos\theta = -\frac{1}{5}

gives

1 - 2\sin\theta\cos\theta = \frac{1}{25}

, which simplifies to

\sin\theta\cos\theta = \frac{12}{25}

. We then find the square of the sum:

(\sin\theta + \cos\theta)^2 = 1 + 2\sin\theta\cos\theta = 1 + \frac{24}{25} = \frac{49}{25}

. Since the angle is in the third quadrant, both trigonometric functions are negative, so we choose the negative root

\sin\theta + \cos\theta = -\frac{7}{5}

. Factoring the sum of cubes gives

\sin^3\theta + \cos^3\theta = (\sin\theta + \cos\theta)(\sin^2\theta - \sin\theta\cos\theta + \cos^2\theta) = (-\frac{7}{5})(1 - \frac{12}{25}) = -\frac{91}{125}

. Multiplying this by 125 yields -91.

Adım Adım Çözüm

1

Square both sides of the equation

\sin\theta - \cos\theta = -\frac{1}{5}

\sin^2\theta - 2\sin\theta\cos\theta + \cos^2\theta = \frac{1}{25} \implies 1 - 2\sin\theta\cos\theta = \frac{1}{25} \implies \sin\theta\cos\theta = \frac{12}{25}

To solve for the product of sine and cosine using the Pythagorean identity

\sin^2\theta + \cos^2\theta = 1

.

2

Determine the value of

\sin\theta + \cos\theta

using the identity

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

(\sin\theta + \cos\theta)^2 = 1 + 2(\frac{12}{25}) = \frac{49}{25} \implies \sin\theta + \cos\theta = -\frac{7}{5}

Because

\theta

is in the third quadrant (

\pi < \theta < \frac{3\pi}{2}

), both

\sin\theta

and

\cos\theta

must be negative, meaning their sum is also negative.

3

Use the sum of cubes factorization to evaluate

\sin^3\theta + \cos^3\theta

\sin^3\theta + \cos^3\theta = (\sin\theta + \cos\theta)(\sin^2\theta - \sin\theta\cos\theta + \cos^2\theta) = (-\frac{7}{5})(1 - \frac{12}{25}) = -\frac{91}{125}

To express the sum of cubes in terms of the known sum and product of sine and cosine.

4

Multiply the evaluated sum of cubes by 125

125 \times (-\frac{91}{125}) = -91

To compute the final value of the requested expression.

Anahtar Kavram

Pythagorean trigonometric identities, quadrant sign analysis, and algebraic factorization of the sum of cubes

Alternatif Yöntem

Instead of applying algebraic identities to find the sum of cubes directly, we can solve for the individual values of

\sin\theta

and

\cos\theta

from the system of equations:

\sin\theta - \cos\theta = -\frac{1}{5}

and

\sin^2\theta + \cos^2\theta = 1

. Substituting

\sin\theta = \cos\theta - \frac{1}{5}

into the second equation yields

2\cos^2\theta - \frac{2}{5}\cos\theta - \frac{24}{25} = 0

, which factors as

(5\cos\theta + 3)(5\cos\theta - 4) = 0

. Since

\theta

is in Quadrant III,

\cos\theta = -\frac{3}{5}

and

\sin\theta = -\frac{4}{5}

. Evaluating

125(\sin^3\theta + \cos^3\theta)

directly with these values gives

125((-\frac{4}{5})^3 + (-\frac{3}{5})^3) = 125(-\frac{64}{125} - \frac{27}{125}) = -91

.

Tahmini Süre:2m 30s

Soru 413Soru →

Chords $AB$ and $CD$ intersect at point $E$ inside a circle. If $AE = 6$ , $EB = 8$ , and the total length of chord $CD$ is $16$ , what is the length of the shorter segment of chord $CD$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The length of the shorter segment of chord

CD

is

4

.

According to the Intersecting Chords Theorem, when two chords intersect inside a circle, the product of the segments of one chord is equal to the product of the segments of the other. For chords

AB

and

CD

intersecting at point

E

, this relationship is expressed as

AE \cdot EB = CE \cdot ED

. Substituting the given values yields

6 \cdot 8 = CE \cdot ED

, so

CE \cdot ED = 48

. Since the total length of chord

CD

is

16

, we can define

CE = x

and

ED = 16 - x

. The equation becomes

x(16 - x) = 48

, which simplifies to the quadratic equation

x^2 - 16x + 48 = 0

. Factoring this equation gives

(x - 12)(x - 4) = 0

, meaning the two segments of chord

CD

have lengths of

12

and

4

. The length of the shorter segment is

4

.

Adım Adım Çözüm

1

State the relationship between intersecting chord segments.

AE \cdot EB = CE \cdot ED

By the Intersecting Chords Theorem, the product of the segments of one chord equals the product of the segments of the other.

2

Substitute the known lengths and define the segments of

CD

using a variable

x

.

6 \cdot 8 = x(16 - x)

, which simplifies to

48 = 16x - x^2

.

We are given

AE = 6

and

EB = 8

. Since the total length of chord

CD

is

16

, if one segment is

x

, the remaining segment must be

16 - x

.

3

Solve the quadratic equation for

x

by factoring.

x^2 - 16x + 48 = 0 \implies (x - 12)(x - 4) = 0

, so

x = 12

or

x = 4

.

Rearranging the equation into standard quadratic form allows us to find the two possible segment lengths.

4

Identify the shorter segment length from the two solutions.

4

The two segment lengths are

12

and

4

. The problem asks for the shorter segment, which is

4

.

Anahtar Kavram

Intersecting Chords Theorem

Soru 414Soru →

In triangle $ABC$ , the length of side $a$ is $5$ centimeters, the length of side $b$ is $8$ centimeters, and the measure of angle $C$ is $60^\circ$ . What is the length, in centimeters, of side $c$ ?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The length of side

c

is

7

centimeters.

Applying the Law of Cosines directly to the given Side-Angle-Side (SAS) triangle yields

c^2 = 5^2 + 8^2 - 2(5)(8) \cos 60^\circ = 25 + 64 - 40 = 49

, which gives

c = 7

.

Adım Adım Çözüm

1

Identify the given values and the appropriate formula.

We are given two sides,

a = 5

and

b = 8

, and the included angle

C = 60^\circ

. To find the opposite side

c

, we use the Law of Cosines:

c^2 = a^2 + b^2 - 2ab \cos C

.

The Law of Cosines relates three sides of a triangle to the cosine of one of its angles, which is applicable for Side-Angle-Side (SAS) configurations.

2

Substitute the known values into the Law of Cosines equation.

c^2 = 5^2 + 8^2 - 2(5)(8) \cos 60^\circ

Plugging the given values into the formula allows us to solve for the unknown side

c

.

3

Evaluate the trigonometric and arithmetic terms.

Since

\cos 60^\circ = 0.5

, we get:

c^2 = 25 + 64 - 80(0.5)

c^2 = 89 - 40

c^2 = 49

Simplifying the expression step-by-step leads to the value of

c^2

.

4

Take the square root of both sides to find the side length.

c = \sqrt{49} = 7

Since side lengths must be positive, the square root of

49

gives the exact length of side

c

.

Anahtar Kavram

Law of Cosines

Soru 415Soru →

In the figure, point $B$ lies on the line segment $AC$ , and segment $BD$ is perpendicular to $AC$ . Triangle $ABD$ is a right triangle with hypotenuse $AD = 8$ and $\angle ADB = 30^\circ$ . If the length of segment $BC$ is $11$ , what is the length of segment $CD$ ?

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

The length of segment

CD

is

13

.

The length of segment

CD

is

13

. Since

BD

is perpendicular to segment

AC

at point

B

,

\triangle ABD

and

\triangle DBC

are both right triangles. In the

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

right triangle

\triangle ABD

, the hypotenuse is

AD = 8

, so the longer leg opposite the

60^\circ

angle is

BD = 4\sqrt{3}

. In right triangle

\triangle DBC

, using the Pythagorean Theorem:

CD^2 = BD^2 + BC^2 = (4\sqrt{3})^2 + 11^2 = 48 + 121 = 169

. Taking the square root gives

CD = 13

.

Adım Adım Çözüm

1

Find the length of the shared perpendicular segment

BD

using the properties of the special

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

right triangle

\triangle ABD

.

BD = 4\sqrt{3}

In a

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

right triangle, the side opposite the

60^\circ

angle is

\frac{\sqrt{3}}{2}

times the hypotenuse.

2

Apply the Pythagorean Theorem to right triangle

\triangle DBC

to calculate the length of hypotenuse

CD

.

CD = 13

The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the legs (

CD^2 = BD^2 + BC^2

).

Anahtar Kavram

Solving for unknown sides in adjacent right triangles by combining special right triangle ratios (

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

) and the Pythagorean Theorem.

Soru 416Soru →

In the standard $(x, y)$ coordinate plane, a parallelogram $ABCD$ has vertices $A(-6, -3)$ , $B(1, -3)$ , $C(6, 2)$ , and $D(-1, 2)$ . What is the length of the diagonal $AC$ ?

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

The correct answer is 13.

The length of the diagonal

AC

is found by calculating the distance between the vertices

A(-6, -3)

and

C(6, 2)

. Using the distance formula:

AC = \sqrt{(6 - (-6))^2 + (2 - (-3))^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13

.

Adım Adım Çözüm

1

Identify the coordinates of the endpoints of diagonal

AC

.

A(-6, -3)

and

C(6, 2)

The diagonal of a polygon connects non-adjacent vertices, so the diagonal

AC

connects vertex

A

to vertex

C

.

2

Use the distance formula to find the length of the segment

AC

.

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

The distance formula calculates the straight-line distance between two points on the coordinate plane.

3

Simplify the expression to find the final integer length.

13

Calculating the squares and summing them gives

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

.

Anahtar Kavram

Calculating the length of a geometric segment on the coordinate plane using the distance formula.

Soru 417Soru →

An angle $\theta$ in standard position measures $-750^\circ$ . Let $\phi$ be the coterminal angle of $\theta$ such that $0 \le \phi < 2\pi$ radians. If $\phi = \frac{a\pi}{b}$ , where $a$ and $b$ are positive integers with no common factors, what is the value of $a + b$ ?

Cevabı ve açıklamayı göster

Cevap: 17

Cevap

17

To find the coterminal angle

\phi

in the range

[0, 2\pi)

radians, we first determine the coterminal angle in degrees by adding multiples of

360^\circ

. Adding

3 \times 360^\circ = 1080^\circ

to

-750^\circ

results in

330^\circ

. We then convert

330^\circ

to radians by multiplying by

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

, which simplifies to

\frac{11\pi}{6}

. Since

11

and

6

are positive integers with no common factors,

a = 11

and

b = 6

. The sum

a + b

is

11 + 6 = 17

.

Adım Adım Çözüm

1

Find the positive coterminal angle of

-750^\circ

within one full rotation.

330^\circ

Adding

1080^\circ

(three full rotations of

360^\circ

) to

-750^\circ

brings the angle within the standard range of

[0^\circ, 360^\circ)

.

2

Convert the angle from degrees to radians.

\frac{11\pi}{6}

radians

Multiplying the degree measure by

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

and simplifying the fraction converts it to radians.

3

Identify the values of

a

and

b

and compute their sum.

17

Comparing

\frac{11\pi}{6}

to

\frac{a\pi}{b}

shows

a = 11

and

b = 6

, which share no common factors. Their sum is

11 + 6 = 17

.

Anahtar Kavram

Finding coterminal angles and converting angle measures between degrees and radians.

Tahmini Süre:1m 30s

Soru 418Soru →

A line graphed on the standard $(x, y)$ coordinate plane has a slope of $0.5$ and passes through the points $(p, 3)$ and $(7, 2p - 1)$ . What is the value of $p$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The value of

p

is

3

.

Applying the slope formula

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

to the points

(p, 3)

and

(7, 2p - 1)

with

m = 0.5

yields the linear equation

0.5 = \frac{2p - 4}{7 - p}

. Cross-multiplying and solving for

p

gives

p = 3

.

Adım Adım Çözüm

1

Set up the slope formula using the given points and slope.

0.5 = \frac{(2p - 1) - 3}{7 - p}

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by the formula

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

.

2

Simplify the numerator of the fraction.

0.5 = \frac{2p - 4}{7 - p}

Combine the constant terms

-1

and

-3

in the numerator.

3

Multiply both sides by the denominator to clear the fraction.

0.5(7 - p) = 2p - 4

Eliminating the denominator allows solving the linear equation.

4

Distribute the

0.5

on the left side of the equation.

3.5 - 0.5p = 2p - 4

Multiply each term inside the parentheses by

0.5

.

5

Isolate the variable

p

by rearranging terms.

2.5p = 7.5

Add

0.5p

to both sides and add

4

to both sides.

6

Divide by the coefficient of

p

to solve for

p

.

p = 3

Dividing

7.5

by

2.5

yields the final value of the parameter.

Anahtar Kavram

Calculating a parameter using the slope formula

Alternatif Yöntem

Instead of using the decimal

0.5

, write it as the fraction

\frac{1}{2}

. The equation becomes

\frac{1}{2} = \frac{2p - 4}{7 - p}

. Cross-multiplying gives

1(7 - p) = 2(2p - 4) \Rightarrow 7 - p = 4p - 8 \Rightarrow 15 = 5p \Rightarrow p = 3

.

Tahmini Süre:1m 30s

Soru 419Soru →

A surveyor is measuring a triangular plot of land, $ABC$ . The distance from point $A$ to point $C$ is $12$ meters, and the distance from point $B$ to point $C$ is $6\sqrt{2}$ meters. If the measure of angle $BAC$ is $30^\circ$ , what is the measure, in degrees, of the acute angle $ABC$ ?

Cevabı ve açıklamayı göster

Cevap: 45

Cevap

The measure of the acute angle

ABC

is

45

degrees.

Applying the Law of Sines yields the relation

\frac{\sin(B)}{12} = \frac{\sin(30^\circ)}{6\sqrt{2}}

. Solving for

\sin(B)

yields

\sin(B) = \frac{\sqrt{2}}{2}

. Because the question specifies that the angle is acute, the measure of the angle is

45^\circ

.

Adım Adım Çözüm

1

Set up the Law of Sines relationship for triangle

ABC

using the side opposite angle

B

(

AC

) and the side opposite angle

A

(

BC

).

\frac{\sin(B)}{AC} = \frac{\sin(A)}{BC}

The Law of Sines states that the ratio of the sine of an angle to the length of its opposite side is constant for all three angles in a triangle.

2

Substitute the given values into the equation:

AC = 12

,

BC = 6\sqrt{2}

, and

A = 30^\circ

.

\frac{\sin(B)}{12} = \frac{\sin(30^\circ)}{6\sqrt{2}}

This allows us to solve for the single unknown variable, the sine of angle

B

.

3

Simplify the expression using the trigonometric value

\sin(30^\circ) = 0.5

and isolate

\sin(B)

.

\sin(B) = \frac{12 \cdot 0.5}{6\sqrt{2}} = \frac{6}{6\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}

Simplifying the fractions helps us identify the standard trigonometric value.

4

Find the acute angle

B

whose sine value is

\frac{\sqrt{2}}{2}

.

B = 45^\circ

The inverse sine of

\frac{\sqrt{2}}{2}

for an acute angle is

45^\circ

.

Anahtar Kavram

Law of Sines

Tahmini Süre:1m 0s

Soru 420Soru →

An angle $\theta$ is positioned in the second quadrant. If the trigonometric expression $\csc\theta - \cot\theta$ is equal to $4$ , what is the value of $17\cos\theta$ ?

Cevabı ve açıklamayı göster

Cevap: -15

Cevap

The value of

17\cos\theta

is

-15

.

The correct value is

-15

. By using the difference of squares on the identity

\csc^2\theta - \cot^2\theta = 1

, we establish that

\csc\theta + \cot\theta = \frac{1}{4}

. Solving this system alongside

\csc\theta - \cot\theta = 4

yields

\csc\theta = \frac{17}{8}

and

\cot\theta = -\frac{15}{8}

. Since

\cos\theta = \frac{\cot\theta}{\csc\theta}

, we find

\cos\theta = -\frac{15}{17}

, which means

17\cos\theta = -15

.

Adım Adım Çözüm

1

Use the Pythagorean identity linking cosecant and cotangent to find the sum of the terms.

\csc\theta + \cot\theta = \frac{1}{4}

Since

\csc^2\theta - \cot^2\theta = 1

, factoring as a difference of squares gives

(\csc\theta - \cot\theta)(\csc\theta + \cot\theta) = 1

. Substituting

\csc\theta - \cot\theta = 4

yields

4(\csc\theta + \cot\theta) = 1

, so

\csc\theta + \cot\theta = \frac{1}{4}

.

2

Solve the system of equations for

\csc\theta

and

\cot\theta

.

\csc\theta = \frac{17}{8}

and

\cot\theta = -\frac{15}{8}

Adding the equations

\csc\theta - \cot\theta = 4

and

\csc\theta + \cot\theta = \frac{1}{4}

gives

2\csc\theta = \frac{17}{4} \implies \csc\theta = \frac{17}{8}

. Subtracting the first equation from the second gives

2\cot\theta = -\frac{15}{4} \implies \cot\theta = -\frac{15}{8}

.

3

Determine the value of

\cos\theta

using reciprocal and quotient identities.

\cos\theta = -\frac{15}{17}

Using the relationship

\cos\theta = \frac{\cot\theta}{\csc\theta}

, we substitute the values to find

\cos\theta = \frac{-15/8}{17/8} = -\frac{15}{17}

.

4

Calculate the final required expression value.

17\cos\theta = -15

Multiplying the calculated value of

\cos\theta

by

17

gives

17 \left(-\frac{15}{17}\right) = -15

.

Anahtar Kavram

Using fundamental Pythagorean, reciprocal, and quotient trigonometric identities to solve systems of equations and evaluate trigonometric expressions.

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Alternatif Yöntem

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Alternatif Yöntem

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Sık Yapılan Hatalar

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