Geometry and Trigonometry Practice Questions — SAT — Page 8

Question 141Question →

A logo design consists of a region bounded by an isosceles trapezoid and a semicircle. The semicircle is attached to the shorter parallel side of the trapezoid such that the diameter of the semicircle is equal to the length of that side, and the semicircle lies entirely outside the trapezoid. The trapezoid has a height of $8$ centimeters, a longer parallel side of $18$ centimeters, and a shorter parallel side of $10$ centimeters. What is the total area, in square centimeters, of the logo?

224 + 12.5\pi

112 + 50\pi

112 + 12.5\pi

112 + 25\pi

Show answer & explanation

Answer: $112 + 12.5\pi$

Answer

The total area of the logo is

112 + 12.5\pi

square centimeters.

The total area of the logo is the sum of the area of the trapezoid and the area of the semicircle. The area of the trapezoid is calculated using the formula

A = \frac{b_1 + b_2}{2}h

, which gives

\frac{18 + 10}{2} \times 8 = 112

square centimeters. The area of the semicircle is half the area of a full circle with a diameter of

10

centimeters (radius of

5

centimeters), which is

\frac{1}{2}\pi r^2 = \frac{1}{2}\pi (5^2) = 12.5\pi

square centimeters. Adding these two areas together gives

112 + 12.5\pi

square centimeters.

Step-by-Step Solution

1

Calculate the area of the trapezoid section of the logo.

The area of the trapezoid is

112

square centimeters.

The area of a trapezoid is given by the formula

A = \frac{b_1 + b_2}{2}h

, where

b_1 = 18

,

b_2 = 10

, and

h = 8

.

2

Calculate the area of the semicircle section of the logo.

The area of the semicircle is

12.5\pi

square centimeters.

The diameter of the semicircle is the shorter base of the trapezoid (

10

centimeters), so the radius is

r = 5

centimeters. The area of a semicircle is half the area of a full circle:

A = \frac{1}{2}\pi r^2 = \frac{1}{2}\pi (5)^2 = 12.5\pi

.

3

Sum the areas of the trapezoid and the semicircle to find the total area.

The total area is

112 + 12.5\pi

square centimeters.

The total area of a composite figure is the sum of the areas of its non-overlapping component parts.

Key Concept

Calculating the total area of a composite figure by decomposing it into standard shapes (a trapezoid and a semicircle) and summing their individual areas.

Estimated Time:1m 30s

Question 142Question →

In triangle $ABC$ , angle $C$ is a right angle. The length of side $AC$ is $15$ and the length of side $BC$ is $8$ . Point $D$ lies on side $AC$ such that $CD = 6$ . A line segment $DE$ is drawn perpendicular to side $AC$ such that point $E$ lies on side $AB$ . What is the length of segment $BE$ ?

7.8

6.8

10.2

9.1

Show answer & explanation

Answer: 6.8

Answer

The correct answer is 6.8.

The length of the hypotenuse

AB

is calculated using the Pythagorean theorem:

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

. The length of segment

AD

is

15 - 6 = 9

. Because segment

DE

is perpendicular to

AC

, the right triangle

ADE

shares angle

A

with right triangle

ACB

, making them similar. Using the ratio of corresponding sides,

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

, we get

\frac{9}{15} = \frac{AE}{17}

, which simplifies to

AE = 10.2

. Finally, subtracting

AE

from the total hypotenuse length gives

BE = 17 - 10.2 = 6.8

.

Step-by-Step Solution

1

Find the length of the hypotenuse

AB

using the Pythagorean theorem.

AB = 17

For a right triangle with legs

AC = 15

and

BC = 8

, the hypotenuse

AB

is given by

\sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17

.

2

Calculate the length of segment

AD

.

AD = 9

Since point

D

lies on side

AC

and the total length of

AC

is

15

, the length of

AD

is

AC - CD = 15 - 6 = 9

.

3

Identify the similarity relationship between triangle

ADE

and triangle

ACB

.

\triangle ADE \sim \triangle ACB

Both triangles share angle

A

, and both have a right angle (

\angle ADE = \angle ACB = 90^\circ

). Therefore, by Angle-Angle similarity, the triangles are similar.

4

Use the similarity ratio to find the length of segment

AE

.

AE = 10.2

The ratio of corresponding sides gives

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

, which simplifies to

\frac{9}{15} = \frac{AE}{17}

, or

0.6 = \frac{AE}{17}

. Solving for

AE

gives

AE = 17 \times 0.6 = 10.2

.

5

Subtract

AE

from the total hypotenuse

AB

to find the length of segment

BE

.

BE = 6.8

The length of segment

BE

is the difference between the total hypotenuse

AB

and the segment

AE

, which is

17 - 10.2 = 6.8

.

Key Concept

Applying the Pythagorean theorem and similar right triangles to find missing lengths on a hypotenuse.

Question 143Question →

In the $xy$ -plane, a circle with radius $r$ , where $r > 1$ , has its center in the first quadrant. The circle is tangent to the line $x = 1$ and tangent to the line $y = 2$ . If the center of the circle lies on the line with equation $y = \frac{4}{3}x$ , what is the value of $r$ ?

Show answer & explanation

Answer: 2

Answer

The radius

r

of the circle is 2.

Since the circle has radius

r

and is tangent to the lines

x = 1

and

y = 2

, its center

(h, k)

lies at a distance of

r

from both lines. This means

h = 1 \pm r

and

k = 2 \pm r

. Because the center is in the first quadrant,

h > 0

and

k > 0

. Given

r > 1

, the choice

h = 1 - r

would make

h

negative, so we must have

h = r + 1

. If

k = 2 - r

, the center is

(r + 1, 2 - r)

, and substituting this into the line

y = \frac{4}{3}x

gives

2 - r = \frac{4}{3}(r + 1) \implies 6 - 3r = 4r + 4 \implies 7r = 2 \implies r = \frac{2}{7}

, which contradicts the condition that

r > 1

. Therefore, we must have

k = r + 2

. Substituting the center

(r + 1, r + 2)

into

y = \frac{4}{3}x

yields

r + 2 = \frac{4}{3}(r + 1) \implies 3(r + 2) = 4(r + 1) \implies 3r + 6 = 4r + 4 \implies r = 2

.

Step-by-Step Solution

1

Set up equations for the center coordinates

(h, k)

in terms of the radius

r

.

|h - 1| = r

and

|k - 2| = r

The distance from the center of a circle to any of its tangent lines is equal to the radius

r

.

2

Determine the correct sign for the absolute value expression of the

x

-coordinate.

h = r + 1

Since the center lies in the first quadrant,

h

must be positive. If

h = 1 - r

, then

r > 1

would imply

h < 0

, which is a contradiction.

3

Determine the correct sign for the absolute value expression of the

y

-coordinate by evaluating both possibilities on the line

y = \frac{4}{3}x

.

k = r + 2

If

k = 2 - r

, then substituting into the line equation gives

r = \frac{2}{7}

, which contradicts the condition that

r > 1

. Thus,

k

must equal

r + 2

.

4

Substitute

(r + 1, r + 2)

into the line equation

y = \frac{4}{3}x

and solve for

r

.

r = 2

Substituting gives

r + 2 = \frac{4}{3}(r + 1) \implies 3(r + 2) = 4(r + 1) \implies 3r + 6 = 4r + 4 \implies r = 2

.

Key Concept

Equations of Circles in the Coordinate Plane

Question 144Question →

Points $A$ , $B$ , and $C$ lie on a circle. The measure of minor arc $AB$ is $110^\circ$ and the measure of minor arc $BC$ is $130^\circ$ . What is the measure of the inscribed angle $\angle ABC$ , in degrees?

120

50

60

55

Show answer & explanation

Answer: 60

Answer

60

The measure of the major arc

ABC

is the sum of the minor arcs

AB

and

BC

, which is

110^\circ + 130^\circ = 240^\circ

. The remaining minor arc

AC

has a measure of

360^\circ - 240^\circ = 120^\circ

. By the inscribed angle theorem, the measure of the inscribed angle

\angle ABC

is half the measure of its intercepted arc, minor arc

AC

. Therefore, the measure of

\angle ABC

is

\frac{120^\circ}{2} = 60^\circ

.

Step-by-Step Solution

1

Calculate the measure of the major arc

ABC

by adding the measures of the two adjacent minor arcs

AB

and

BC

.

The measure of arc

ABC

is

110^\circ + 130^\circ = 240^\circ

.

Since points

A

,

B

, and

C

are in order on the circle, the major arc connecting

A

and

C

through

B

is the sum of the arcs

AB

and

BC

.

2

Find the measure of the remaining minor arc

AC

.

The measure of minor arc

AC

is

360^\circ - 240^\circ = 120^\circ

.

A full circle measures

360^\circ

. Subtracting the major arc

ABC

from

360^\circ

yields the measure of the minor arc

AC

.

3

Apply the inscribed angle theorem to find the measure of angle

ABC

.

The measure of angle

ABC

is

\frac{120^\circ}{2} = 60^\circ

.

The inscribed angle theorem states that the measure of an inscribed angle is half the measure of its intercepted arc. Angle

ABC

intercepts the minor arc

AC

.

Key Concept

The measure of an inscribed angle is half the measure of its intercepted arc.

Question 145Question →

Triangle $ABC$ is similar to triangle $DEF$ , where the ratio of the length of side $AB$ to the length of side $DE$ is $3$ to $5$ . If the area of triangle $ABC$ is $18$ , what is the area of triangle $DEF$ ?

Show answer & explanation

Answer: 50

Answer

The area of triangle

DEF

is

50

.

Since triangle

ABC

is similar to triangle

DEF

, the ratio of their areas is the square of the ratio of their corresponding side lengths. The ratio of side

AB

to side

DE

is

3/5

, so the ratio of the area of triangle

ABC

to the area of triangle

DEF

is

(3/5)^2 = 9/25

. Given that the area of triangle

ABC

is

18

, we can set up the proportion

18 / x = 9 / 25

, where

x

represents the area of triangle

DEF

. Solving for

x

gives

x = 18 \times (25/9) = 2 \times 25 = 50

.

Step-by-Step Solution

1

Determine the ratio of the areas of the two similar triangles using their side length ratio.

The ratio of the area of triangle

ABC

to the area of triangle

DEF

is

(3/5)^2 = 9/25

.

For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths.

2

Set up a proportion to solve for the unknown area of triangle

DEF

.

\frac{18}{x} = \frac{9}{25}

, where

x

is the area of triangle

DEF

.

We equate the ratio of the actual areas to the theoretical ratio of areas derived from the side lengths.

3

Solve the proportion for

x

.

x = 18 \times \frac{25}{9} = 2 \times 25 = 50

.

Multiplying both sides by the reciprocal isolates the variable and yields the area.

Key Concept

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

Question 146Question →

In a circle with center $O$ , segment $PT$ is tangent to the circle at point $T$ . The distance from point $P$ to the center of the circle is $25$ . If the radius of the circle is $7$ , what is the length of segment $PT$ ?

Show answer & explanation

Answer: 24

Answer

The length of segment

PT

is 24.

Because segment

PT

is tangent to the circle at point

T

, the radius

OT

is perpendicular to

PT

. This forms a right triangle

OTP

where the right angle is at vertex

T

, the legs are

OT = 7

and

PT

, and the hypotenuse is the segment from the center to the external point

OP = 25

. By the Pythagorean theorem,

OT^2 + PT^2 = OP^2

. Substituting the known lengths yields

7^2 + PT^2 = 25^2

, which simplifies to

49 + PT^2 = 625

. Subtracting

49

from both sides gives

PT^2 = 576

. Taking the square root of both sides results in

PT = 24

.

Step-by-Step Solution

1

Identify the relationship between the radius and the tangent line at the point of tangency.

The radius

OT

is perpendicular to the tangent segment

PT

, making triangle

OTP

a right triangle with a

90^\circ

angle at vertex

T

.

A tangent line to a circle is always perpendicular to the radius drawn to the point of tangency.

2

Set up the Pythagorean theorem for the right triangle

OTP

.

OT^2 + PT^2 = OP^2

In any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

3

Substitute the given values

OT = 7

and

OP = 25

into the equation and solve for the length of

PT

.

PT = 24

Substituting values gives

7^2 + PT^2 = 25^2 \implies 49 + PT^2 = 625 \implies PT^2 = 576 \implies PT = \sqrt{576} = 24

.

Key Concept

A line tangent to a circle is perpendicular to the radius at the point of tangency, allowing the use of the Pythagorean theorem to find unknown lengths in the resulting right triangle.

Question 147Question →

In a circle, chords $WY$ and $XZ$ intersect at point $P$ . The measure of minor arc $WX$ is $55^\circ$ and the measure of minor arc $YZ$ is $105^\circ$ . What is the measure, in degrees, of angle $WPX$ ?

Show answer & explanation

Answer: 80

Answer

80

The correct answer is 80. According to the intersecting chords angle theorem, when two chords intersect inside a circle, the measure of the angle they form is half the sum of the measures of the intercepted arcs. Here, angle

WPX

and its vertical angle intercept minor arcs

WX

and

YZ

. Therefore, the measure of angle

WPX

is

\frac{55^\circ + 105^\circ}{2} = \frac{160^\circ}{2} = 80^\circ

.

Step-by-Step Solution

1

Identify the geometric relationship for angles formed by intersecting chords inside a circle.

The measure of

\angle WPX

is equal to half the sum of the measures of its intercepted arc

WX

and the intercepted arc of its vertical angle, arc

YZ

.

By the intersecting chords angle theorem, the angle formed by two intersecting chords inside a circle is half the sum of the measures of the intercepted arcs.

2

Sum the measures of the intercepted arcs.

55^\circ + 105^\circ = 160^\circ

The measures of minor arcs

WX

and

YZ

are given as

55^\circ

and

105^\circ

respectively.

3

Divide the sum of the arc measures by 2.

80

Halving the sum of the arc measures (

160^\circ

) yields the measure of the angle:

\frac{160^\circ}{2} = 80^\circ

.

Key Concept

Intersecting Chords Angle Theorem

Question 148Question →

A designer has a rectangular piece of fabric that measures $12$ inches by $18$ inches. The designer cuts out two identical right triangular pieces from the corners along one of the $12$ -inch sides. Each right triangular piece has a leg of length $4$ inches along the $12$ -inch side and a leg of length $x$ inches along the $18$ -inch side. If the area of the remaining piece of fabric is $180$ square inches, what is the value of $x$ ?

Show answer & explanation

Answer: 9

Answer

9

The correct answer is 9. The original area of the rectangular fabric is

12 \times 18 = 216

square inches. Two identical right triangles with leg lengths of

4

inches and

x

inches are cut out. The area of each triangle is

\frac{1}{2} \times 4 \times x = 2x

square inches. The total area of the two triangles is

2 \times 2x = 4x

square inches. Subtracting this from the original area gives the remaining area:

216 - 4x = 180

. Solving for

x

gives

4x = 36

, which simplifies to

x = 9

.

Step-by-Step Solution

1

Calculate the area of the original rectangular piece of fabric.

216 square inches

To find the initial area before any modifications are made, using the formula

\text{Area} = \text{length} \times \text{width}

.

2

Find the total area of the two cut-out right triangles in terms of

x

.

4x

square inches

Each right triangle has legs of

4

and

x

, so its area is

\frac{1}{2}(4)(x) = 2x

. The total area of two such identical triangles is

2(2x) = 4x

.

3

Set up an equation using the remaining area of the fabric.

216 - 4x = 180

The remaining area of

180

square inches is equal to the original area of

216

square inches minus the total area of the two cut-out triangles, which is

4x

.

4

Solve the equation for

x

.

x = 9

Isolating the variable

x

by subtracting

216

from both sides and then dividing by

-4

yields

x = 9

.

Key Concept

Area of composite shapes (rectangles and triangles)

Question 149Question →

In a circle with center $O$ , chord $AB$ has a length of $12$ . The perpendicular distance from center $O$ to chord $AB$ is $8$ . If the area of the circle is $k\pi$ , what is the value of $k$ ?

Show answer & explanation

Answer: 100

Answer

100

A perpendicular from the center of a circle to a chord bisects that chord. For a chord of length

12

, the perpendicular split creates two segments of length

6

. Drawing a radius from the center to one of the chord's endpoints forms a right triangle with legs of

6

and

8

. By the Pythagorean theorem, the hypotenuse (which is the radius

r

) satisfies

r^2 = 6^2 + 8^2 = 100

. The area of the circle is

\pi r^2 = 100\pi

. Thus, the coefficient

k

is

100

.

Step-by-Step Solution

1

Determine the length of half of the chord.

6

A line segment drawn perpendicular from the center of a circle to a chord bisects the chord. Therefore, the distance from the midpoint of the chord to either endpoint is

12 / 2 = 6

.

2

Use the Pythagorean theorem to calculate the square of the radius.

r^2 = 100

The radius, half of the chord, and the perpendicular distance form a right-angled triangle. According to the Pythagorean theorem, the hypotenuse squared (

r^2

) is the sum of the squares of the legs:

r^2 = 6^2 + 8^2 = 36 + 64 = 100

.

3

Calculate the area of the circle in terms of

\pi

and identify the value of

k

.

k = 100

The formula for the area of a circle is

\pi r^2

. Since

r^2 = 100

, the area is

100\pi

. Comparing this to

k\pi

, we find that

k = 100

.

Key Concept

Perpendicular bisector of a circle chord and right triangle properties

Estimated Time:1m 30s

Question 150Question →

A circle with center $O$ has a radius of $8$ . Points $A$ and $B$ lie on the circle such that the length of the minor arc $AB$ is $5\pi$ . What is the area of the major sector $AOB$ ?

20\pi

36\pi

44\pi

11\pi

Show answer & explanation

Answer: $44\pi$

Answer

The area of the major sector is

44\pi

.

The area of the major sector is found by scaling the total area of the circle,

64\pi

, by the ratio of the major arc length to the circumference. The circumference is

2\pi(8) = 16\pi

. Since the minor arc length is

5\pi

, the major arc length is

16\pi - 5\pi = 11\pi

. The ratio of the major arc to the circumference is

\frac{11\pi}{16\pi} = \frac{11}{16}

. Multiplying this ratio by the total area of the circle yields

\frac{11}{16} \times 64\pi = 44\pi

.

Step-by-Step Solution

1

Calculate the circumference of the circle.

16\pi

The circumference formula is

C = 2\pi r

. Given

r = 8

, the circumference is

2\pi(8) = 16\pi

.

2

Find the length of the major arc

AB

.

11\pi

The length of the major arc is the total circumference minus the length of the minor arc:

16\pi - 5\pi = 11\pi

.

3

Calculate the total area of the circle.

64\pi

The area formula for a circle is

A = \pi r^2

. Given

r = 8

, the total area is

\pi(8^2) = 64\pi

.

4

Determine the area of the major sector by scaling the total area.

44\pi

The major sector's area is proportional to the fraction of the circle represented by the major arc:

\frac{11\pi}{16\pi} \times 64\pi = \frac{11}{16} \times 64\pi = 44\pi

.

Key Concept

Calculating sector area using arc length and total circle area relationships.

Estimated Time:1m 30s

Question 151Question →

A property line is in the shape of a trapezoid with parallel sides of length $20$ meters and $30$ meters, and a height of $12$ meters. A second property is geometrically similar to the first, where each linear dimension of the second property is $1.5$ times the corresponding dimension of the first property. What is the area, in square meters, of the second property?

450

675

900

1,350

Show answer & explanation

Answer: 675

Answer

675

The area of the first property is calculated using the trapezoid area formula:

A = \frac{1}{2}(b_1 + b_2)h

. Substituting the given values gives

A = \frac{1}{2}(20 + 30)(12) = 300

square meters. Since the second property is similar to the first with a linear scale factor of

1.5

, its area is scaled by

1.5^2 = 2.25

. Therefore, the area of the second property is

300 \times 2.25 = 675

square meters.

Step-by-Step Solution

1

Calculate the area of the first trapezoidal property.

300

square meters

Using the trapezoid area formula

A = \frac{1}{2}(b_1 + b_2)h

, we find

A = \frac{1}{2}(20 + 30)(12) = 300

.

2

Determine the area scale factor for the similar property.

2.25

Since the second property is similar to the first with a linear scale factor of

1.5

, its area scales by the square of this factor:

(1.5)^2 = 2.25

.

3

Calculate the area of the second property.

675

square meters

Multiplying the original area by the area scale factor yields

300 \times 2.25 = 675

.

Key Concept

Area of similar two-dimensional shapes scales by the square of the linear scale factor.

Question 152Question →

In right triangle $PQR$ , the measure of angle $Q$ is $90^\circ$ . If $\sin(P) = \frac{5}{13}$ and the length of side $QR$ is $15$ , what is the length of side $PQ$ ?

Show answer & explanation

Answer: 36

Answer

The length of side

PQ

is

36

.

By definition,

\sin(P) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{QR}{PR}

. Given that

\sin(P) = \frac{5}{13}

and

QR = 15

, we set up the equation

\frac{5}{13} = \frac{15}{PR}

and solve for the hypotenuse

PR

, giving

PR = 39

. Using the Pythagorean theorem,

PQ^2 + QR^2 = PR^2

, we substitute the known values:

PQ^2 + 15^2 = 39^2 \implies PQ^2 + 225 = 1521 \implies PQ^2 = 1296

. Taking the square root of both sides gives

PQ = 36

. Alternatively, recognizing that the sides of the triangle form a

5

-

12

-

13

Pythagorean triple scaled by a factor of

3

(since

QR = 5 \times 3 = 15

and

PR = 13 \times 3 = 39

), the remaining leg

PQ

must be

12 \times 3 = 36

.

Step-by-Step Solution

1

Set up the sine ratio for angle

P

to find the length of the hypotenuse

PR

.

PR = 39

Since

\sin(P)

is the ratio of the opposite side (

QR

) to the hypotenuse (

PR

), we can solve the equation

\frac{5}{13} = \frac{15}{PR}

to find that

PR = 39

.

2

Apply the Pythagorean theorem to solve for the length of side

PQ

.

PQ = 36

In right triangle

PQR

, the relationship between the sides is

PQ^2 + QR^2 = PR^2

. Substituting

QR = 15

and

PR = 39

gives

PQ^2 + 15^2 = 39^2

, which simplifies to

PQ^2 = 1296

, so

PQ = 36

.

Key Concept

Using trigonometric ratios to find side lengths of right triangles followed by the Pythagorean theorem.

Alternative Method

Recognize that the triangle's sides must be a multiple of the common

5

-

12

-

13

Pythagorean triple. Since the opposite side is

15

(

5 \times 3

) and the hypotenuse is

39

(

13 \times 3

), the scaling factor is

3

, meaning the adjacent side

PQ

is

12 \times 3 = 36

.

Estimated Time:1m 30s

Question 153Question →

In a circle with center $O$ , the length of minor arc $AB$ is $3\pi$ and the area of sector $AOB$ is $18\pi$ . What is the circumference of the circle?

12\pi

18\pi

24\pi

36\pi

Show answer & explanation

Answer: $24\pi$

Answer

The circumference of the circle is

24\pi

.

The correct answer is the option containing

24\pi

. By using the relationship

A = \frac{1}{2}rs

, where

A

is the sector area,

r

is the radius, and

s

is the arc length, we substitute the given values to get

18\pi = \frac{1}{2}r(3\pi)

. Solving for the radius yields

r = 12

. Substituting this radius into the circumference formula

C = 2\pi r

gives

2\pi(12) = 24\pi

.

Step-by-Step Solution

1

Write the formulas for arc length

s

and sector area

A

in terms of radius

r

and central angle

\theta

in radians.

s = r\theta = 3\pi

and

A = \frac{1}{2}r^2\theta = 18\pi

.

This sets up the system of equations using the given geometric properties.

2

Express the sector area formula in terms of arc length by substituting

s = r\theta

into

A = \frac{1}{2}r(r\theta)

.

A = \frac{1}{2}rs

, which becomes

18\pi = \frac{1}{2}r(3\pi)

.

This simplifies the relationship to a single equation with one variable,

r

.

3

Solve the equation

18\pi = 1.5\pi r

for the radius

r

.

r = 12

.

Finding the radius is necessary to calculate the circumference of the circle.

4

Substitute

r = 12

into the circumference formula

C = 2\pi r

.

C = 2\pi(12) = 24\pi

.

This provides the final circumference value requested by the question.

Key Concept

Relationship between arc length, sector area, and circumference in circle geometry

Question 154Question →

Line segments $AC$ and $BD$ intersect at point $E$ such that segment $AB$ is parallel to segment $CD$ . If the length of $AE$ is $5$ , the length of $CE$ is $10$ , the length of $BE$ is $x - 2$ , and the length of $DE$ is $x + 4$ , what is the value of $x$ ?

Show answer & explanation

Answer: 8

Answer

8

Since segment

AB

is parallel to segment

CD

, the alternate interior angles

\angle EAB

and

\angle ECD

are congruent, and vertical angles

\angle AEB

and

\angle CED

are congruent. By the Angle-Angle (AA) similarity theorem, triangle

ABE

is similar to triangle

CDE

. The ratio of the lengths of corresponding sides is equal, so

\frac{AE}{CE} = \frac{BE}{DE}

. Substituting the given lengths gives

\frac{5}{10} = \frac{x - 2}{x + 4}

. Simplifying the left side to

\frac{1}{2}

and cross-multiplying gives

x + 4 = 2(x - 2)

, which expands to

x + 4 = 2x - 4

. Solving for

x

yields

x = 8

.

Step-by-Step Solution

1

Establish the similarity of triangles

ABE

and

CDE

.

\triangle ABE \sim \triangle CDE

Since segment

AB

is parallel to segment

CD

, alternate interior angles

\angle EAB

and

\angle ECD

are congruent, and vertical angles

\angle AEB

and

\angle CED

are congruent. Thus, the triangles are similar by AA similarity.

2

Set up a proportion using the ratio of corresponding sides.

\frac{AE}{CE} = \frac{BE}{DE}

In similar triangles, the ratio of corresponding side lengths is constant.

3

Substitute the given algebraic expressions and segment lengths into the proportion.

\frac{5}{10} = \frac{x - 2}{x + 4}

The given values are

AE = 5

,

CE = 10

,

BE = x - 2

, and

DE = x + 4

.

4

Simplify the fraction and solve the linear equation for

x

.

x = 8

Simplifying

\frac{5}{10}

yields

\frac{1}{2}

. Cross-multiplying gives

1 \cdot (x + 4) = 2 \cdot (x - 2)

, which simplifies to

x + 4 = 2x - 4

. Subtracting

x

from both sides and adding

4

to both sides gives

x = 8

.

Key Concept

Triangle similarity criteria (specifically AA similarity) and using proportions of corresponding sides in similar triangles to solve for unknown variables.

Question 155Question →

In triangle $ABC$ , the side lengths are $AB = AC = 5$ and $BC = 6$ . Point $D$ is the midpoint of side $BC$ , and point $E$ lies on side $AB$ such that segment $DE$ is perpendicular to side $AB$ . What is the length of segment $DE$ ?

Show answer & explanation

Answer: 2.4

Answer

The length of segment

DE

is

2.4

(or the equivalent fraction

12/5

).

The correct answer is

2.4

(or

12/5

). In the isosceles triangle

ABC

with

AB = AC = 5

, the median

AD

to the base

BC

is also an altitude. Since

D

is the midpoint of

BC

, we have

BD = 3

. Applying the Pythagorean theorem to right triangle

ABD

gives

AD = \sqrt{5^2 - 3^2} = 4

. Since

DE

is perpendicular to

AB

, triangle

AED

is a right triangle that shares angle

A

with right triangle

ADB

. Therefore, triangle

AED

is similar to triangle

ADB

. The ratio of the opposite side to the hypotenuse in both triangles must be equal:

\frac{DE}{BD} = \frac{AD}{AB}

, which gives

\frac{DE}{3} = \frac{4}{5}

, or

DE = 2.4

.

Step-by-Step Solution

1

Determine the properties of the altitude

AD

in the isosceles triangle

ABC

.

AD

is perpendicular to

BC

, and

BD = 3

.

In an isosceles triangle, the median to the base is also the altitude to the base. Since

D

is the midpoint of

BC

,

BD = \frac{BC}{2} = \frac{6}{2} = 3

, and

\angle ADB = 90^\circ

.

2

Calculate the length of segment

AD

using the Pythagorean theorem in right triangle

ABD

.

AD = 4

Applying the Pythagorean theorem to right triangle

ABD

gives

AD^2 + BD^2 = AB^2

. Substituting the known lengths yields

AD^2 + 3^2 = 5^2

, which simplifies to

AD^2 = 25 - 9 = 16

, so

AD = 4

.

3

Find the length of segment

DE

using triangle similarity.

DE = 2.4

Since segment

DE

is perpendicular to side

AB

,

\angle AED = 90^\circ

. The right triangles

AED

and

ADB

share the angle at

A

, so they are similar by AA similarity (

\triangle AED \sim \triangle ADB

). This allows us to set up the ratio of corresponding sides:

\frac{DE}{BD} = \frac{AD}{AB}

. Substituting the values gives

\frac{DE}{3} = \frac{4}{5}

, which results in

DE = \frac{12}{5} = 2.4

.

Key Concept

Properties of isosceles triangles, the Pythagorean theorem, and right triangle similarity theorems.

Alternative Method

Alternatively, the length of

DE

can be found using the area of right triangle

ABD

. The area of triangle

ABD

is

\frac{1}{2} \times BD \times AD = \frac{1}{2} \times 3 \times 4 = 6

. The area can also be expressed using the hypotenuse

AB

as the base and

DE

as the height:

\text{Area} = \frac{1}{2} \times AB \times DE = \frac{1}{2} \times 5 \times DE

. Setting these equal gives

\frac{5}{2} DE = 6

, which yields

DE = 2.4

.

Estimated Time:1m 30s

Question 156Question →

In triangle $ABC$ , point $D$ lies on side $AB$ and point $E$ lies on side $AC$ such that segment $DE$ is parallel to segment $BC$ . The length of segment $AD$ is $2x + 1$ , the length of segment $DB$ is $x + 1$ , the length of segment $AE$ is $10$ , and the length of segment $EC$ is $6$ . What is the length of segment $AB$ ?

Show answer & explanation

Answer: 8

Answer

The length of segment

AB

is 8.

By the Triangle Proportionality Theorem, since segment

DE

is parallel to segment

BC

, the segments on the transversal sides are proportional:

\frac{AD}{DB} = \frac{AE}{EC}

. Substituting the given expressions and values yields

\frac{2x + 1}{x + 1} = \frac{10}{6}

. Simplifying the fraction on the right side to

\frac{5}{3}

and cross-multiplying gives

3(2x + 1) = 5(x + 1)

. Solving this equation yields

x = 2

. The length of segment

AB

is the sum of

AD

and

DB

, which is

(2x + 1) + (x + 1) = 3x + 2

. Substituting

x = 2

gives

AB = 8

.

Step-by-Step Solution

1

Set up the proportion using the Triangle Proportionality Theorem.

\frac{AD}{DB} = \frac{AE}{EC}

Since segment

DE

is parallel to segment

BC

, it divides the sides of triangle

ABC

proportionally.

2

Substitute the given values and simplify the constant ratio.

\frac{2x + 1}{x + 1} = \frac{5}{3}

The length of segment

AE

is 10 and

EC

is 6, so

\frac{AE}{EC} = \frac{10}{6} = \frac{5}{3}

.

3

Cross-multiply and solve the linear equation for

x

.

x = 2

Cross-multiplying gives

3(2x + 1) = 5(x + 1)

, which simplifies to

6x + 3 = 5x + 5

. Subtracting

5x

and 3 from both sides yields

x = 2

.

4

Calculate the total length of segment

AB

.

AB = 8

The total length

AB

is the sum of

AD

and

DB

. Thus,

AB = (2x + 1) + (x + 1) = 3x + 2

. Substituting

x = 2

gives

3(2) + 2 = 8

.

Key Concept

Triangle Proportionality Theorem and Similar Triangles

Alternative Method

Instead of using the Triangle Proportionality Theorem directly, we can use the similarity of triangles

ADE

and

ABC

. Since

DE \parallel BC

, we have

\triangle ADE \sim \triangle ABC

by AA similarity. This gives the ratio of corresponding side lengths:

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

. Substituting the expressions yields

\frac{2x + 1}{3x + 2} = \frac{10}{16} = \frac{5}{8}

. Cross-multiplying gives

8(2x + 1) = 5(3x + 2) \implies 16x + 8 = 15x + 10 \implies x = 2

. Then,

AB = 3x + 2 = 3(2) + 2 = 8

.

Estimated Time:1m 30s

Question 157Question →

In the $xy$ -plane, the circle defined by the equation $x^2 + y^2 - 10x + 12y = k$ has a radius of $9$ , where $k$ is a constant. What is the value of $k$ ?

-52

70

20

142

Show answer & explanation

Answer: 20

Answer

20

To find the value of

k

, we first rewrite the given equation in the standard form of a circle's equation,

(x - h)^2 + (y - k_{center})^2 = r^2

. Starting with

x^2 + y^2 - 10x + 12y = k

, we complete the square for the

x

-terms by adding

(\frac{-10}{2})^2 = 25

to both sides, and for the

y

-terms by adding

(\frac{12}{2})^2 = 36

to both sides. This yields

(x^2 - 10x + 25) + (y^2 + 12y + 36) = k + 25 + 36

, which simplifies to

(x - 5)^2 + (y + 6)^2 = k + 61

. In this standard form, the right-hand side represents the square of the radius, so

r^2 = k + 61

. Given that the radius is

9

, we have

r^2 = 9^2 = 81

. Setting

k + 61 = 81

and subtracting

61

from both sides gives

k = 20

.

Step-by-Step Solution

1

Write the given equation of the circle.

x^2 + y^2 - 10x + 12y = k

To establish the starting equation before completing the square.

2

Complete the square for the

x

and

y

terms by adding

(\frac{-10}{2})^2 = 25

and

(\frac{12}{2})^2 = 36

to both sides.

(x^2 - 10x + 25) + (y^2 + 12y + 36) = k + 25 + 36

To express the quadratic expressions as perfect squares.

3

Rewrite the equation in standard form.

(x - 5)^2 + (y + 6)^2 = k + 61

To match the standard equation of a circle,

(x-h)^2 + (y-k_{center})^2 = r^2

, where the right-hand side represents the square of the radius.

4

Equate the constant term on the right-hand side to

r^2

using the given radius

r = 9

, and solve for

k

.

k + 61 = 9^2 \implies k + 61 = 81 \implies k = 20

To calculate the value of the constant

k

that satisfies the radius requirement.

Key Concept

Equations of Circles in the Coordinate Plane

Question 158Question →

In a circle with center $O$ and radius $5$ , points $A$ , $B$ , and $C$ lie on the circle. If the measure of the inscribed angle $\angle ABC$ is $\frac{2\pi}{5}$ radians, what is the length of minor arc $AC$ ?

\pi

2\pi

4\pi

8\pi

Show answer & explanation

Answer: $4\pi$

Answer

4\pi

According to the Inscribed Angle Theorem, the measure of a central angle is twice the measure of an inscribed angle that subtends the same arc. Given that the inscribed angle

\angle ABC

has a measure of

\frac{2\pi}{5}

radians, the corresponding central angle

\angle AOC

has a measure of

2 \times \frac{2\pi}{5} = \frac{4\pi}{5}

radians. Using the formula for arc length,

s = r\theta

, where

r = 5

is the radius and

\theta = \frac{4\pi}{5}

is the central angle in radians, the length of minor arc

AC

is

5 \times \frac{4\pi}{5} = 4\pi

.

Step-by-Step Solution

1

Find the measure of the central angle

\angle AOC

that subtends the same minor arc

AC

as the inscribed angle

\angle ABC

.

The measure of central angle

\angle AOC

is

2 \times \frac{2\pi}{5} = \frac{4\pi}{5}

radians.

By the Inscribed Angle Theorem, the measure of a central angle subtending an arc is twice the measure of any inscribed angle subtending the same arc.

2

Calculate the length of minor arc

AC

using the formula

s = r\theta

.

The arc length is

s = 5 \times \frac{4\pi}{5} = 4\pi

.

The formula for the arc length of a circle is

s = r\theta

, where

r

is the radius and

\theta

is the central angle measure in radians.

Key Concept

Inscribed Angle Theorem and Arc Length in Radians

Question 159Question →

Points $A$ , $B$ , and $C$ lie on a circle with center $O$ . The length of the minor arc $AC$ is $\frac{4}{9}$ of the circumference of the circle. What is the measure, in degrees, of the inscribed angle $\angle ABC$ ?

Show answer & explanation

Answer: 80

Answer

The measure of the inscribed angle is 80 degrees.

The correct answer is 80. The minor arc

AC

constitutes

\frac{4}{9}

of the circle's circumference, which corresponds to an arc measure of

\frac{4}{9} \times 360^\circ = 160^\circ

. By the Inscribed Angle Theorem, the measure of the inscribed angle

\angle ABC

is half the measure of the intercepted arc, which is

\frac{1}{2} \times 160^\circ = 80^\circ

.

Step-by-Step Solution

1

Determine the degree measure of the minor arc

AC

.

The measure of minor arc

AC

is

160^\circ

.

Since a full circle has a circumference corresponding to

360^\circ

, minor arc

AC

has a degree measure of

\frac{4}{9} \times 360^\circ = 160^\circ

.

2

Calculate the measure of the inscribed angle

\angle ABC

.

The measure of

\angle ABC

is

80^\circ

.

According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of the arc it intercepts. Thus, the measure of

\angle ABC

is

\frac{1}{2} \times 160^\circ = 80^\circ

.

Key Concept

Inscribed Angle Theorem and Arc Measure

Question 160Question →

In the figure below, $AD$ is the angle bisector of $\angle BAC$ in triangle $ABC$ . The length of segment $AB$ is $2x$ , the length of segment $AC$ is $3x-3$ , the length of segment $BD$ is $6$ , and the length of segment $CD$ is $8$ . What is the value of $x$ ?

2

3

6

9

Show answer & explanation

Answer: 9

Answer

The value of

x

is 9.

According to the Angle Bisector Theorem, the bisector of an angle in a triangle divides the opposite side into segments that are proportional to the adjacent sides. This gives the proportion

\frac{AB}{AC} = \frac{BD}{CD}

. Substituting the given values, we get

\frac{2x}{3x-3} = \frac{6}{8}

. Simplifying the right side to

\frac{3}{4}

and cross-multiplying yields

2x(4) = 3(3x-3)

, which simplifies to

8x = 9x - 9

. Solving for

x

gives

x = 9

.

Step-by-Step Solution

1

Apply the Angle Bisector Theorem, which states that an angle bisector in a triangle divides the opposite side into two segments that are proportional to the other two sides.

\frac{AB}{AC} = \frac{BD}{CD}

Because

AD

is the angle bisector of

\angle BAC

, it splits

BC

at

D

proportionally to the adjacent sides

AB

and

AC

.

2

Substitute the given side lengths into the proportion.

\frac{2x}{3x-3} = \frac{6}{8}

To set up the algebraic equation for

x

using the given lengths

AB = 2x

,

AC = 3x-3

,

BD = 6

, and

CD = 8

.

3

Simplify the ratio on the right side of the equation and cross-multiply to solve for

x

.

8x = 9x - 9

, which simplifies to

x = 9

.

First simplify

\frac{6}{8}

to

\frac{3}{4}

, then cross-multiply:

2x(4) = 3(3x-3)

, which gives

8x = 9x - 9

. Subtracting

8x

and adding

9

yields

x = 9

.

Key Concept

Angle Bisector Theorem in Triangles

Geometry and Trigonometry

Answer

Step-by-Step Solution

Key Concept

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