Plane Geometry Practice Questions — ACT — Page 7

Question 121Question →

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

Show answer & explanation

Answer: 4

Answer

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

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

Step-by-Step Solution

1

Determine the vertical height of the campsite and the tower.

The campsite

C

is

12

miles north of the trailhead

A

's east-west line. Since the tower

T

is located due west of

C

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

T

is also

12

miles.

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

2

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

Let

D

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

T

. A right triangle

\triangle ADT

is formed where the vertical leg

DT = 12

miles, the hypotenuse

AT = 13

miles, and the horizontal leg is

AD

. Using the Pythagorean theorem:

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

miles.

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

3

Calculate the horizontal distance between the campsite and the tower.

The campsite

C

is

9

miles east of the trailhead

A

's north-south line, and the tower

T

is

5

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

9 - 5 = 4

miles.

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

Key Concept

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

Estimated Time:1m 30s

Question 122Question →

A windshield wiper of length $15\text{ inches}$ sweeps through a central angle of $120^\circ$ across a windshield. What is the area, in square inches, of the region swept by the wiper?

10\pi

75\pi

112.5\pi

150\pi

225\pi

Show answer & explanation

Answer: $75\pi$

Answer

75\pi

square inches

The area of the region swept by the wiper is the area of a circle sector with a radius of

15\text{ inches}

and a central angle of

120^\circ

. The formula for the area of a sector is

A = \frac{\theta}{360} \times \pi r^2

. Substituting

15

for the radius and

120

for the angle gives

A = \frac{120}{360} \times \pi (15)^2 = \frac{1}{3} \times 225\pi = 75\pi

square inches.

Step-by-Step Solution

1

Determine the formula for the area of a sector of a circle.

A = \frac{\theta}{360^\circ} \times \pi r^2

The area of a sector is a fraction of the total area of the circle, where the fraction is determined by the central angle

\theta

divided by the total degrees in a circle (

360^\circ

).

2

Substitute the given values into the sector area formula, using a radius of

15

and a central angle of

120^\circ

.

A = \frac{120^\circ}{360^\circ} \times \pi (15)^2

The windshield wiper length represents the radius

r = 15\text{ inches}

, and the sweep angle represents the central angle

\theta = 120^\circ

.

3

Simplify the expression to find the final area.

A = \frac{1}{3} \times 225\pi = 75\pi

Reducing the fraction

\frac{120}{360}

to

\frac{1}{3}

and squaring

15

to get

225

yields the area of

75\pi

square inches.

Key Concept

The area of a circle sector is found by multiplying the total circle area,

\pi r^2

, by the ratio of the central angle to the total degree measure of a circle,

\frac{\theta}{360^\circ}

.

Question 123Question →

For any convex quadrilateral $ABCD$ , is the statement that $AB^2 + CD^2 = BC^2 + DA^2$ if and only if the diagonals $AC$ and $BD$ are perpendicular true or false?

True

False

Show answer & explanation

Answer: True

Answer

The statement is true because the sum of the squares of the lengths of opposite sides in a convex quadrilateral is equal if and only if its diagonals intersect at right angles.

The statement is true because the equality of the sums of the squares of opposite sides is mathematically equivalent to the diagonals being perpendicular in any convex quadrilateral.

Step-by-Step Solution

1

Define the intersection point of the diagonals

AC

and

BD

as

P

and the angle of intersection as

\theta

.

Four triangles are formed:

\triangle APB

,

\triangle BPC

,

\triangle CPD

, and

\triangle DPA

, with angles at

P

being

\theta

and

180^\circ - \theta

.

This establishes a geometric frame of reference to relate side lengths to diagonal segments.

2

Apply the Law of Cosines to express the square of each side length in terms of the diagonal segments

AP

,

BP

,

CP

, and

DP

.

AB^2 = AP^2 + BP^2 - 2(AP)(BP)\cos\theta

,

CD^2 = CP^2 + DP^2 - 2(CP)(DP)\cos\theta

,

BC^2 = BP^2 + CP^2 + 2(BP)(CP)\cos\theta

, and

DA^2 = DP^2 + AP^2 + 2(DP)(AP)\cos\theta

.

This links the boundary side lengths of the quadrilateral to its internal diagonals.

3

Sum the squares of the opposite sides and compute their difference:

(AB^2 + CD^2) - (BC^2 + DA^2)

.

(AB^2 + CD^2) - (BC^2 + DA^2) = -2(AP\cdot BP + CP\cdot DP + BP\cdot CP + DP\cdot AP)\cos\theta

.

This algebraic combination isolates the term involving the angle of intersection.

4

Factor the coefficient of

-2\cos\theta

and simplify the relation.

(AB^2 + CD^2) - (BC^2 + DA^2) = -2(AP + CP)(BP + DP)\cos\theta = -2(AC)(BD)\cos\theta

.

Factoring groups the individual segments into the full lengths of the diagonals

AC

and

BD

.

5

Analyze the condition for the difference to be zero.

AB^2 + CD^2 = BC^2 + DA^2 \iff \cos\theta = 0 \iff \theta = 90^\circ

.

Since the lengths

AC

and

BD

must be positive, the difference is zero if and only if the diagonals are perpendicular.

Key Concept

Orthodiagonal quadrilateral properties and diagonal relations

Question 124Question →

In the trapezoid $ABCD$ , bases $AB$ and $CD$ are parallel, and side $AD$ is perpendicular to base $AB$ . The length of $AB$ is $10$ inches, the length of $CD$ is $4$ inches, and the measure of angle $B$ is $60^\circ$ . What is the perimeter, in inches, of the trapezoid?

20 + 3\sqrt{3}

20 + 6\sqrt{3}

26 + 6\sqrt{3}

14 + 6\sqrt{3}

32\sqrt{3}

Show answer & explanation

Answer: $26 + 6\sqrt{3}$

Answer

The perimeter of the trapezoid is

26 + 6\sqrt{3}

inches.

Drawing an altitude from vertex

C

to base

AB

splits the trapezoid into a rectangle

AECD

and a right triangle

\triangle CEB

. Since

AE = CD = 4

, the leg

EB = 10 - 4 = 6

. The right triangle

\triangle CEB

is a

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

triangle where

EB

is the leg opposite the

30^\circ

angle. The hypotenuse

BC = 2 \times 6 = 12

and the height

CE = AD = 6\sqrt{3}

. Summing all four outer sides of the trapezoid (

10 + 12 + 4 + 6\sqrt{3}

) yields the correct perimeter of

26 + 6\sqrt{3}

inches.

Step-by-Step Solution

1

Draw an altitude from vertex

C

perpendicular to base

AB

at point

E

.

A rectangle

AECD

and a right triangle

\triangle CEB

are formed, with

AE = CD = 4

inches and

CE = AD

.

To break down the trapezoid into a rectangle and a right triangle so we can find the unknown side lengths.

2

Find the length of segment

EB

.

EB = AB - AE = 10 - 4 = 6

inches.

To find the length of the base of the right triangle

\triangle CEB

.

3

Use the properties of a

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

right triangle to determine the lengths of sides

CE

(which is

AD

) and

BC

.

Since

\angle B = 60^\circ

is opposite to

CE

, and

EB = 6

is the shorter leg adjacent to

60^\circ

, the hypotenuse is

BC = 2 \times 6 = 12

inches and the longer leg is

CE = AD = 6\sqrt{3}

inches.

To compute the remaining unknown outer side lengths of the trapezoid.

4

Sum the four outer sides of the trapezoid to find the perimeter.

Perimeter =

AB + BC + CD + DA = 10 + 12 + 4 + 6\sqrt{3} = 26 + 6\sqrt{3}

inches.

The perimeter is the total boundary length of the shape.

Key Concept

The perimeter of a right trapezoid can be found by drawing an altitude to create a rectangle and a

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

special right triangle, then determining the missing side lengths using special right triangle ratios.

Estimated Time:1m 30s

Question 125Question →

An irregular convex heptagon (7-sided polygon) has three interior angles that each measure $140^\circ$ . The remaining four interior angles have measures in the ratio $2:3:3:4$ . What is the measure of the largest interior angle of this heptagon?

80^\circ

120^\circ

140^\circ

160^\circ

300^\circ

Show answer & explanation

Answer: $160^\circ$

Answer

160^\circ

The total sum of the interior angles of a convex heptagon is

(7-2) \times 180^\circ = 900^\circ

. Subtracting the three angles that each measure

140^\circ

(

3 \times 140^\circ = 420^\circ

) leaves

480^\circ

for the remaining four angles. Using the ratio

2:3:3:4

, we represent the angles as

2x

,

3x

,

3x

, and

4x

, giving the equation

2x + 3x + 3x + 4x = 480^\circ

, which simplifies to

12x = 480^\circ

and yields

x = 40^\circ

. The largest of these four angles is

4x = 4(40^\circ) = 160^\circ

. Since

160^\circ

is larger than

140^\circ

, the largest interior angle of the heptagon is

160^\circ

.

Step-by-Step Solution

1

Calculate the sum of the interior angles of a heptagon.

900^\circ

The sum of the interior angles of a convex polygon with

n

sides is given by

(n-2) \times 180^\circ

. For a heptagon (

n=7

), the sum is

(7-2) \times 180^\circ = 5 \times 180^\circ = 900^\circ

.

2

Find the sum of the three known interior angles.

420^\circ

Since three interior angles each measure

140^\circ

, their sum is

3 \times 140^\circ = 420^\circ

.

3

Calculate the sum of the remaining four interior angles.

480^\circ

Subtracting the sum of the three known angles from the total interior sum gives

900^\circ - 420^\circ = 480^\circ

.

4

Set up and solve an equation for the remaining four angles using the given ratio.

x = 40^\circ

Let the measures of the remaining four angles be

2x

,

3x

,

3x

, and

4x

. Their sum is

2x + 3x + 3x + 4x = 12x

. Setting this equal to the remaining sum gives

12x = 480^\circ

, which simplifies to

x = 40^\circ

.

5

Determine the measures of the remaining angles and identify the largest angle.

160^\circ

The remaining angles measure

2(40^\circ) = 80^\circ

,

3(40^\circ) = 120^\circ

,

3(40^\circ) = 120^\circ

, and

4(40^\circ) = 160^\circ

. The largest of these is

160^\circ

. Comparing this with the three

140^\circ

angles, the largest interior angle of the entire heptagon is

160^\circ

.

Key Concept

Interior angle sum of polygons and ratio division

Question 126Question →

A circle is inscribed inside a sector of a larger circle. The larger circle has a radius of $18\text{ inches}$ and the sector has a central angle of $60^\circ$ , as shown in the figure. What is the area, in square inches, of the region that is inside the sector but outside the inscribed circle?

27\pi

45\pi

18\pi

72\pi

288\pi

Show answer & explanation

Answer: $18\pi$

Answer

The correct area is

18\pi

square inches.

The correct answer is

18\pi

square inches. First, the sector area is calculated as

\frac{60}{360} \times \pi \times 18^2 = 54\pi

. Second, using the right triangle formed by the sector's center, the inscribed circle's center, and the point of tangency, we set up

\sin(30^\circ) = \frac{r}{18-r}

. Solving for

r

gives

r = 6

, so the area of the inscribed circle is

\pi \times 6^2 = 36\pi

. The difference between the two areas is

54\pi - 36\pi = 18\pi

.

Step-by-Step Solution

1

Calculate the area of the

60^\circ

sector of the larger circle.

Sector Area =

54\pi

square inches

The sector has a radius of

R = 18

and a central angle of

60^\circ

. The sector's area is a fraction of the total circle's area:

\text{Area}_{\text{sector}} = \frac{60}{360} \times \pi R^2 = \frac{1}{6} \times \pi \times 18^2 = \frac{324\pi}{6} = 54\pi

.

2

Determine the radius

r

of the inscribed circle using trigonometry.

Inscribed Radius

r = 6

inches

The center of the inscribed circle,

I

, lies on the angle bisector of the sector. The line segment from the center of the sector

O

to

I

bisects the

60^\circ

angle, forming a

30^\circ

angle. The distance from

O

to the outer boundary of the sector is

R = 18

, and the distance from

I

to the boundary is

r

, so the hypotenuse

OI = 18 - r

. Drawing a perpendicular from

I

to one of the straight edges of the sector creates a right triangle with opposite side

r

(the radius) and hypotenuse

18 - r

. Applying the sine ratio:

\sin(30^\circ) = \frac{r}{18 - r}

. Since

\sin(30^\circ) = 0.5

, we solve

0.5 = \frac{r}{18 - r} \Rightarrow 18 - r = 2r \Rightarrow 3r = 18 \Rightarrow r = 6

.

3

Calculate the area of the inscribed circle.

Inscribed Circle Area =

36\pi

square inches

The area of the inscribed circle with radius

r = 6

is

\text{Area}_{\text{circle}} = \pi r^2 = \pi \times 6^2 = 36\pi

.

4

Subtract the area of the inscribed circle from the area of the sector.

Remaining Area =

18\pi

square inches

The area of the region inside the sector but outside the circle is found by subtracting the inscribed circle area from the sector area:

54\pi - 36\pi = 18\pi

.

Key Concept

Using trigonometry on angle bisectors to determine the radius of a circle inscribed inside a sector, and applying sector and circle area formulas.

Estimated Time:3m 0s

Question 127Question →

A convex polygon has $n$ sides. The sum of the measures of $n-1$ of its interior angles is $2030^\circ$ . What is the measure, in degrees, of the remaining interior angle?

Show answer & explanation

Answer: 130

Answer

The measure of the remaining interior angle is 130 degrees.

The sum of the interior angles of a convex polygon must be a multiple of

180^\circ

. The multiple of

180^\circ

immediately greater than

2030^\circ

is

2160^\circ

(which is

12 \times 180^\circ

). The difference between the total sum and the sum of the

n-1

angles is

2160^\circ - 2030^\circ = 130^\circ

. Since

130^\circ

is less than

180^\circ

, this is a valid interior angle for a convex polygon.

Step-by-Step Solution

1

Write the formula for the total sum of the interior angles of a convex polygon.

The sum of the interior angles of a polygon with

n

sides is

(n-2) \times 180^\circ

.

This formula relates the number of sides to the total sum of the interior angles.

2

Apply the properties of convex polygons to set up an inequality for the total sum.

Since the remaining interior angle must be greater than

0^\circ

and less than

180^\circ

, the total sum of all

n

angles must satisfy

2030^\circ < (n-2) \times 180^\circ < 2030^\circ + 180^\circ

, which simplifies to

2030^\circ < (n-2) \times 180^\circ < 2210^\circ

.

By definition, every interior angle of a convex polygon is strictly less than

180^\circ

.

3

Solve the inequality to find the integer value of

n-2

.

Dividing the entire inequality by

180^\circ

gives

11.28 < n-2 < 12.28

. Since the number of sides

n

must be an integer,

n-2

must be the integer

12

.

A polygon must have a whole number of sides.

4

Calculate the total sum of all interior angles and find the remaining angle.

The total sum is

12 \times 180^\circ = 2160^\circ

. Subtracting the sum of the other

n-1

angles gives the remaining angle:

2160^\circ - 2030^\circ = 130^\circ

.

The difference between the total sum of all interior angles and the sum of the

n-1

angles is the measure of the final angle.

Key Concept

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, where each interior angle is strictly less than

180^\circ

.

Question 128Question →

A dog is tied to a post in the center of a flat, grassy yard with a leash that is $12\text{ feet}$ long. If the dog walks along an arc formed by a central angle of $\frac{\pi}{3}$ radians, what is the length, in feet, of the path the dog travels?

2\pi

4\pi

8\pi

12\pi

24\pi

Show answer & explanation

Answer: $4\pi$

Answer

The correct option is the one stating

4\pi

.

The length of the path traveled by the dog is the arc length of a circle with a radius of

12

feet and a central angle of

\frac{\pi}{3}

radians. Using the radian arc length formula,

s = r\theta

, substituting

r = 12

and

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

yields

s = 12 \times \frac{\pi}{3} = 4\pi

feet.

Step-by-Step Solution

1

Identify the given values from the problem statement.

The radius of the circular path is

r = 12

feet, and the central angle is

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

radians.

These parameters are required to calculate the arc length of the path.

2

Recall the formula for the arc length of a circle when the angle is in radians.

The formula is

s = r\theta

.

Since the angle is given in radians, the arc length is directly the product of the radius and the angle.

3

Substitute the values into the formula and calculate the result.

s = 12 \times \frac{\pi}{3} = 4\pi

.

Multiplying the radius by the angle in radians gives the length of the path traveled.

Key Concept

Calculating the arc length of a circle when the central angle is measured in radians using the formula

s = r\theta

.

Question 129Question →

In the standard $(x, y)$ coordinate plane, quadrilateral $ABCD$ is a kite that is not a rhombus, with $AB = BC$ and $AD = CD$ . The vertices $A$ and $C$ are located at $(2, 5)$ and $(8, 13)$ , respectively. If the vertex $B$ is located at $(9, 6)$ , which of the following could be the coordinates of vertex $D$ ?

(1, 12)

(-1, 17)

(-3, 15)

(-3, 3)

(-5, 10)

Show answer & explanation

Answer: $(-3, 15)$

Answer

The coordinates

(-3, 15)

The correct answer is the coordinates

(-3, 15)

. The diagonals of a kite are perpendicular, and the diagonal

BD

perpendicularly bisects the diagonal

AC

. The midpoint of

AC

is

M(5, 9)

and the slope of

AC

is

\frac{4}{3}

. Therefore, the line containing diagonal

BD

must pass through

M(5, 9)

with a perpendicular slope of

-\frac{3}{4}

. The equation of this line is

3x + 4y = 51

. Testing the coordinates

(-3, 15)

shows that the point lies on this line. Since the distance from

(-3, 15)

to

M(5, 9)

is

10

units while the distance from

B(9, 6)

to

M(5, 9)

is

5

units, the diagonals do not bisect each other, confirming the kite is not a rhombus.

Step-by-Step Solution

1

Find the midpoint

M

of diagonal

AC

.

M = \left(\frac{2 + 8}{2}, \frac{5 + 13}{2}\right) = (5, 9)

In a kite where

AB = BC

and

AD = CD

, the diagonal

BD

perpendicularly bisects the diagonal

AC

. Therefore, the line containing

BD

must pass through the midpoint of

AC

.

2

Calculate the slope of diagonal

AC

.

m_{AC} = \frac{13 - 5}{8 - 2} = \frac{8}{6} = \frac{4}{3}

The slope is needed to find the perpendicular slope of the line containing diagonal

BD

.

3

Determine the slope of the line containing diagonal

BD

.

m_{BD} = -\frac{1}{m_{AC}} = -\frac{3}{4}

Since the diagonals of a kite are perpendicular, the slope of the line containing diagonal

BD

is the negative reciprocal of the slope of diagonal

AC

.

4

Write the equation of the line containing diagonal

BD

and test the coordinates of vertex

D

.

y - 9 = -\frac{3}{4}(x - 5) \Rightarrow 3x + 4y = 51

Using the point-slope formula with the midpoint

M(5, 9)

and slope

-\frac{3}{4}

gives the line equation. Testing the correct option

(-3, 15)

gives

3(-3) + 4(15) = -9 + 60 = 51

, which lies on this line. Furthermore, the distance from

(-3, 15)

to

M(5, 9)

is

\sqrt{(-3-5)^2 + (15-9)^2} = 10

, which is different from the distance of

5

between

(9, 6)

and

M(5, 9)

, ensuring the kite is not a rhombus.

Key Concept

Diagonals of a kite are perpendicular, and the diagonal connecting the vertices between the unequal adjacent sides perpendicularly bisects the other diagonal.

Question 130Question →

A vertical flagpole is secured by two straight guide wires anchored to the flat ground on opposite sides of the pole. The first guide wire is $13$ feet long and its anchor is $5$ feet from the base of the pole. The second guide wire is anchored such that it makes a $30^\circ$ angle of elevation with the ground. If both guide wires are attached to the flagpole at the same height, what is the length, in feet, of the second guide wire?

Show answer & explanation

Answer: 24

Answer

The length of the second guide wire is

24

feet.

The correct answer is

24

. First, the height of the attachment point is found using the Pythagorean theorem:

h = \sqrt{13^2 - 5^2} = 12

feet. Since the second wire makes a

30^\circ

angle of elevation with the ground, it forms a 30-60-90 right triangle where the flagpole height of

12

feet is the leg opposite the

30^\circ

angle. The length of the wire is the hypotenuse of this triangle, which is twice the length of the opposite leg:

2 \times 12 = 24

feet.

Step-by-Step Solution

1

Use the Pythagorean theorem to calculate the height of the flagpole where the guide wires are attached.

The flagpole height is

12

feet.

The first guide wire, the flagpole, and the ground form a right triangle with a hypotenuse of

13

feet and a horizontal leg of

5

feet. Thus,

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

.

2

Apply the properties of a 30-60-90 special right triangle to find the length of the second guide wire.

The length of the second guide wire is

24

feet.

The second wire forms a 30-60-90 right triangle with the flagpole and the ground. The angle of elevation is

30^\circ

, which means the side opposite this angle is the vertical height of the flagpole (

12

feet). In a 30-60-90 triangle, the hypotenuse (the wire length) is twice the length of the shorter leg (opposite the

30^\circ

angle), so the length is

2 \times 12 = 24

.

Key Concept

Applying the Pythagorean Theorem and the ratio properties of 30-60-90 special right triangles to solve multi-step geometry problems.

Question 131Question →

A goat is tethered to one of the outer corners of a flat, rectangular shed that measures $6\text{ meters}$ by $8\text{ meters}$ . The tether is $10\text{ meters}$ long. Assuming the goat remains outside the shed, what is the total area, in square meters, of the region the goat can graze?

75\pi

83\pi

80\pi

90\pi

100\pi

Show answer & explanation

Answer: $80\pi$

Answer

The correct grazing area is

80\pi

square meters.

The total grazing area is the sum of three sectors: a sector of radius

10\text{ meters}

with central angle

270^\circ

(area

75\pi

), a sector of radius

4\text{ meters}

with central angle

90^\circ

(area

4\pi

), and a sector of radius

2\text{ meters}

with central angle

90^\circ

(area

\pi

). Adding these yields

80\pi

square meters.

Step-by-Step Solution

1

Determine the area of the main sector.

The main sector has a radius of

10\text{ meters}

and a central angle of

360^\circ - 90^\circ = 270^\circ

. The area is

\frac{270}{360} \times \pi \times 10^2 = 75\pi

square meters.

The corner of the rectangular shed blocks

90^\circ

of a full circle, leaving a

270^\circ

sector.

2

Determine the area of the sector at the corner adjacent to the

6

-meter side.

The tether wraps around the corner, leaving a remaining length of

10 - 6 = 4\text{ meters}

. It sweeps through an angle of

90^\circ

. The area is

\frac{90}{360} \times \pi \times 4^2 = 4\pi

square meters.

When the tether extends past the adjacent corner along the

6

-meter side, the pivot point becomes that corner and the tether length decreases by the side length.

3

Determine the area of the sector at the corner adjacent to the

8

-meter side.

The tether wraps around the corner, leaving a remaining length of

10 - 8 = 2\text{ meters}

. It sweeps through an angle of

90^\circ

. The area is

\frac{90}{360} \times \pi \times 2^2 = \pi

square meters.

When the tether extends past the adjacent corner along the

8

-meter side, the pivot point becomes that corner and the tether length decreases by the side length.

4

Check for overlap and calculate the total grazing area.

The total area is

75\pi + 4\pi + \pi = 80\pi

square meters.

The two smaller sectors are at different corners of the rectangular shed and do not overlap. Summing the three sector areas gives the total grazing region.

Key Concept

Calculating sector areas by determining the correct radii and central angles based on geometric constraints.

Question 132Question →

A circle has a radius of $12\text{ centimeters}$ . What is the length, in centimeters, of the arc intercepted by a central angle of $30^\circ$ ?

\pi

2\pi

12\pi

24\pi

2

Show answer & explanation

Answer: $2\pi$

Answer

2\pi

centimeters

To find the arc length, multiply the total circumference of the circle,

2\pi r

, by the fraction of the circle represented by the central angle,

\frac{\theta}{360^\circ}

. With a radius of

12

centimeters and a central angle of

30^\circ

, this calculation yields

2\pi (12) \times \frac{30}{360} = 24\pi \times \frac{1}{12} = 2\pi

centimeters.

Step-by-Step Solution

1

Identify the formula for arc length:

s = 2\pi r \left(\frac{\theta}{360^\circ}\right)

, where

r

is the radius and

\theta

is the central angle in degrees.

Formula established:

s = 2\pi r \left(\frac{\theta}{360^\circ}\right)

Arc length is the fraction of the total circumference determined by the central angle.

2

Substitute the given radius

r = 12\text{ cm}

and central angle

\theta = 30^\circ

into the formula.

Equation set up:

s = 2\pi (12) \left(\frac{30}{360}\right)

This sets up the specific calculation for the given circle.

3

Simplify the expression to find the final arc length.

s = 24\pi \left(\frac{1}{12}\right) = 2\pi

Simplification yields the exact length of the arc in terms of

\pi

.

Key Concept

The length of an arc is proportional to the fraction of the circle's circumference represented by the central angle.

Estimated Time:1m 0s

Question 133Question →

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 $BP$ ?

Show answer & explanation

Answer: 17

Answer

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

.

Step-by-Step Solution

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

.

Key Concept

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

Estimated Time:2m 0s

Question 134Question →

A circular garden plot has a radius of $10\text{ meters}$ . A sector of the garden with a central angle of $72^\circ$ is planted with roses. What is the area, in square meters, of the sector planted with roses?

4\pi

40\pi

20\pi

72\pi

100\pi

Show answer & explanation

Answer: $20\pi$

Answer

The correct answer is

20\pi

, representing the area of the sector in square meters.

The area of a sector of a circle is calculated using the formula

\text{Area} = \frac{\theta}{360^\circ} \pi r^2

. Substituting

\theta = 72^\circ

and

r = 10\text{ meters}

gives

\frac{72^\circ}{360^\circ} \pi (10)^2 = \frac{1}{5} \cdot 100\pi = 20\pi

.

Step-by-Step Solution

1

Identify the formula for the area of a sector.

\text{Area} = \frac{\theta}{360^\circ} \pi r^2

The area of a sector is a proportional fraction of the total area of the circle.

2

Substitute the given values into the formula.

\text{Area} = \frac{72^\circ}{360^\circ} \pi (10)^2

The central angle is

72^\circ

and the radius is

10\text{ meters}

.

3

Simplify the fraction and the squared term.

\text{Area} = \frac{1}{5} \pi (100)

72/360

simplifies to

1/5

and

10^2 = 100

.

4

Perform the final multiplication.

20\pi

One-fifth of

100

is

20

.

Key Concept

Calculating the area of a circle sector given the radius and the central angle in degrees.

Estimated Time:1m 0s

Question 135Question →

For a certain convex polygon, the ratio of the sum of the interior angles to the sum of the exterior angles is $5:1$ . How many sides does this polygon have?

7

8

10

12

14

Show answer & explanation

Answer: 12

Answer

The correct answer is 12 sides.

The sum of the interior angles of any convex polygon with

n

sides is given by

(n-2) \times 180^\circ

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

360^\circ

. Since the ratio of the sum of the interior angles to the sum of the exterior angles is

5:1

, we can set up the equation:

\frac{(n-2) \times 180^\circ}{360^\circ} = \frac{5}{1}

. Simplifying the fraction on the left gives

\frac{n-2}{2} = 5

. Multiplying both sides by 2 gives

n - 2 = 10

, and adding 2 to both sides gives

n = 12

. Therefore, the polygon has 12 sides.

Step-by-Step Solution

1

Write the formulas for the sum of the interior angles and the sum of the exterior angles of a convex polygon.

The sum of the interior angles is

(n-2) \times 180^\circ

, where

n

is the number of sides. The sum of the exterior angles is always

360^\circ

for any convex polygon.

These formulas represent the components of the given ratio.

2

Set up the ratio equation using the given information.

\frac{(n-2) \times 180^\circ}{360^\circ} = \frac{5}{1}

The problem states that the ratio of the sum of the interior angles to the sum of the exterior angles is

5:1

.

3

Simplify the equation and solve for the number of sides,

n

.

\frac{n-2}{2} = 5 \implies n-2 = 10 \implies n = 12

Simplifying

\frac{180}{360}

to

\frac{1}{2}

makes it easier to solve the algebraic equation for

n

.

Key Concept

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, while the sum of the exterior angles is always

360^\circ

.

Estimated Time:1m 30s

Question 136Question →

A right triangle has a hypotenuse of length $13\text{ inches}$ . One of the legs is $7\text{ inches}$ longer than the other leg. What is the length, in inches, of the shorter leg?

3.0

5.0

6.0

8.6

11.0

Show answer & explanation

Answer: 5.0

Answer

The length of the shorter leg is 5.0 inches.

The correct answer is the option representing 5.0. By setting the shorter leg as

x

, the longer leg is

x + 7

. Applying the Pythagorean theorem gives

x^2 + (x + 7)^2 = 13^2

, which simplifies to

2x^2 + 14x - 120 = 0

. Factoring the divided equation

x^2 + 7x - 60 = 0

gives

(x - 5)(x + 12) = 0

. Since length must be positive, the shorter leg is 5.0 inches.

Step-by-Step Solution

1

Define variables for the side lengths of the right triangle based on the problem statement.

Let the length of the shorter leg be

x

inches. The length of the longer leg is

x + 7

inches, and the hypotenuse is

13

inches.

This translates the verbal descriptions into mathematical expressions using a single variable.

2

Apply the Pythagorean theorem to set up an equation relating the side lengths.

x^2 + (x + 7)^2 = 13^2

For any right triangle, the sum of the squares of the legs equals the square of the hypotenuse.

3

Expand the squared binomial and simplify the equation into standard quadratic form.

x^2 + (x^2 + 14x + 49) = 169 \Rightarrow 2x^2 + 14x - 120 = 0

Expanding allows us to group like terms and solve for the variable

x

.

4

Divide the quadratic equation by 2 and factor the resulting expression.

x^2 + 7x - 60 = 0 \Rightarrow (x + 12)(x - 5) = 0

Factoring is the most efficient way to find the roots of this quadratic equation.

5

Solve for

x

and select the mathematically and physically valid solution.

x = 5

or

x = -12

. Since a side length must be positive,

x = 5

inches.

Lengths in geometry must be positive, so we discard the negative root.

Key Concept

Applying the Pythagorean theorem to solve for unknown side lengths of a right triangle given algebraic relationships between the sides.

Estimated Time:1m 30s

Question 137Question →

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?

Show answer & explanation

Answer: 142

Answer

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

.

Step-by-Step Solution

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

.

Key Concept

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

Question 138Question →

In the standard $(x, y)$ coordinate plane, a quadrilateral $ABCD$ is an isosceles trapezoid with parallel bases $AB$ and $CD$ . The coordinates of three of the vertices are $A(-15, -20)$ , $B(15, 20)$ , and $C(-7, 24)$ . If $ABCD$ is NOT a parallelogram, what are the coordinates of the fourth vertex, $D$ ?

(-37, -16)

(-25, 0)

(-7, -24)

(7, -24)

(23, 64)

Show answer & explanation

Answer: $(-25, 0)$

Answer

(-25, 0)

The correct answer is

(-25, 0)

. First, we find the slope of the parallel bases

AB

and

CD

to be

\frac{4}{3}

, which gives the equation of the line containing

CD

as

4x - 3y + 100 = 0

. Since the trapezoid is isosceles, the leg lengths are equal, meaning

AD^2 = BC^2 = 500

. Substituting the line equation into the distance equation yields two potential coordinates for

D

:

(-25, 0)

and

(-37, -16)

. Since the problem specifies that

ABCD

is not a parallelogram, we eliminate

(-37, -16)

(which makes

AD \parallel BC

) to conclude that

D

must be

(-25, 0)

.

Step-by-Step Solution

1

Calculate the slope of the base

AB

.

The slope of

AB

is

\frac{20 - (-20)}{15 - (-15)} = \frac{40}{30} = \frac{4}{3}

.

Since

AB \parallel CD

, the line containing base

CD

must also have a slope of

\frac{4}{3}

.

2

Set up the equation for the line containing

CD

.

Using the point-slope form with

C(-7, 24)

and slope

\frac{4}{3}

, the equation of the line is

y - 24 = \frac{4}{3}(x + 7) \implies 4x - 3y + 100 = 0

.

The fourth vertex

D(x, y)

must lie on this line, so its coordinates satisfy

x = \frac{3y - 100}{4}

.

3

Set up the distance equation for the equal leg lengths.

The square of the leg length

BC^2 = (-7 - 15)^2 + (24 - 20)^2 = (-22)^2 + 4^2 = 484 + 16 = 500

. Since

AD = BC

, the distance equation is

(x + 15)^2 + (y + 20)^2 = 500

.

In an isosceles trapezoid, the non-parallel sides (legs)

AD

and

BC

must have equal lengths.

4

Solve the system of equations for the coordinates of

D

.

Substitute

x = \frac{3y - 100}{4}

into the distance equation:

(\frac{3y - 40}{4})^2 + (y + 20)^2 = 500 \implies 25y^2 + 400y = 0

. This yields

y = 0

or

y = -16

. The corresponding coordinates are

D_1(-25, 0)

and

D_2(-37, -16)

.

Substituting the linear relationship into the quadratic distance equation gives the two mathematically possible locations for

D

.

5

Verify which solution satisfies the non-parallelogram constraint.

For

D_2(-37, -16)

, the slope of

AD

is

\frac{-16 - (-20)}{-37 - (-15)} = -\frac{2}{11}

, which is equal to the slope of

BC

. This makes

ABCD

a parallelogram. For

D_1(-25, 0)

, the slope of

AD

is

-2 \neq -\frac{2}{11}

, which forms a trapezoid.

The problem states that

ABCD

is not a parallelogram, so

D

must be

(-25, 0)

.

Key Concept

Identifying vertices of an isosceles trapezoid using coordinate geometry, slopes of parallel lines, distance formula, and distinguishing a trapezoid from a parallelogram.

Question 139Question →

A circular archery target has a radius of $12\text{ inches}$ . A sector of this target has a central angle of $150^\circ$ . What is the area, in square inches, of this sector?

10\pi

15\pi

60\pi

120\pi

240\pi

Show answer & explanation

Answer: $60\pi$

Answer

The correct area of the sector is

60\pi

square inches.

To find the area of a sector, first calculate the total area of the circle, which is

\pi r^2 = \pi (12)^2 = 144\pi

square inches. Then, multiply this total area by the fraction of the circle that the sector represents:

\frac{150^\circ}{360^\circ} = \frac{5}{12}

. Calculating the product gives

\frac{5}{12} \times 144\pi = 60\pi

square inches.

Step-by-Step Solution

1

Find the total area of the circular archery target using the area formula

A = \pi r^2

with a radius of

12\text{ inches}

.

The total area of the circle is

\pi (12)^2 = 144\pi

square inches.

The area of a sector is a fractional part of the circle's total area.

2

Calculate the fraction of the circle represented by a central angle of

150^\circ

.

The fraction is

\frac{150^\circ}{360^\circ} = \frac{5}{12}

.

A complete circle has a central angle of

360^\circ

.

3

Multiply the total area of the circle by the fraction of the circle to determine the sector area.

The sector area is

\frac{5}{12} \times 144\pi = 60\pi

square inches.

Applying the fraction to the total area yields the area of the sector.

Key Concept

Calculating the area of a circle sector using the formula

A = \frac{\theta}{360^\circ} \pi r^2

.

Estimated Time:1m 0s

Question 140Question →

A circular sector with radius $R$ and central angle $\theta$ (measured in radians) has the same area and the same perimeter as a square with side length $s$ . What is the value of $\theta$ ?

2

4

8

\frac{1}{\pi}

2\pi

Show answer & explanation

Answer: 2

Answer

The value of theta must be 2

The correct value is 2. By equating the sector's area and perimeter to the square's area and perimeter, we set up a system of equations:

\frac{1}{2} R^2 \theta = s^2

and

2R + R\theta = 4s

. Solving for

s

from the area equation gives

s = R\sqrt{\theta/2}

. Substituting this into the perimeter equation and dividing by

R

yields

2 + \theta = 2\sqrt{2\theta}

. Squaring both sides and simplifying leads to the quadratic equation

\theta^2 - 4\theta + 4 = 0

, which has a single solution of

\theta = 2

radians.

Step-by-Step Solution

1

Write down the equations for the area and perimeter of both shapes.

For the circular sector: Area =

\frac{1}{2}R^2\theta

, Perimeter =

2R + R\theta

(where

\theta

is in radians). For the square: Area =

s^2

, Perimeter =

4s

.

These are the standard geometric formulas for a sector in radians and a square.

2

Set the corresponding areas and perimeters equal to each other to form a system of equations.

Equation 1:

\frac{1}{2} R^2 \theta = s^2

Equation 2:

2R + R\theta = 4s

The problem statement specifies that the two shapes have equal areas and equal perimeters.

3

Solve Equation 1 for the side length

s

of the square.

s = R \sqrt{\frac{\theta}{2}}

This allows us to substitute

s

in Equation 2 and solve for

\theta

in terms of

R

.

4

Substitute the expression for

s

into Equation 2 and simplify.

2R + R\theta = 4\left(R \sqrt{\frac{\theta}{2}}\right) \implies 2 + \theta = 4\sqrt{\frac{\theta}{2}}

Since the radius

R

is a positive length, we can divide both sides of the equation by

R

.

5

Solve the simplified equation for

\theta

by squaring both sides.

2 + \theta = 2\sqrt{2\theta} \implies (2+\theta)^2 = (2\sqrt{2\theta})^2 \implies 4 + 4\theta + \theta^2 = 8\theta \implies \theta^2 - 4\theta + 4 = 0 \implies (\theta - 2)^2 = 0 \implies \theta = 2

Squaring both sides and setting the quadratic equation to zero yields a perfect square trinomial with a single real solution.

Key Concept

Relating the area and perimeter of a circular sector using radian measures to those of a square.

Estimated Time:3m 0s

Plane Geometry

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