Plane Geometry Alıştırma Soruları — ACT — Sayfa 10

Soru 181Soru →

In right triangle $PQR$ , the measure of $\angle PQR$ is $90^\circ$ . Segment $QS$ is an altitude drawn perpendicular to hypotenuse $PR$ with point $S$ lying on $PR$ . If $PQ = 15$ centimeters and $PS = 9$ centimeters, what is the length, in centimeters, of hypotenuse $PR$ ?

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

The length of hypotenuse

PR

is 25 centimeters.

Using the leg-hypotenuse geometric mean theorem for right triangles (

PQ^2 = PS \cdot PR

), substituting

PQ = 15

and

PS = 9

yields

15^2 = 9 \cdot PR \implies 225 = 9 \cdot PR

, solving directly to

PR = 25

cm.

Adım Adım Çözüm

1

Calculate the length of altitude

QS

using the Pythagorean theorem on right triangle

\triangle PQS

QS = \sqrt{15^2 - 9^2} = \sqrt{225 - 81} = \sqrt{144} = 12

cm

Altitude

QS

forms right angle

\angle PSQ = 90^\circ

, making

\triangle PQS

a right triangle.

2

Determine the length of hypotenuse segment

SR

using the geometric mean relationship

QS^2 = PS \cdot SR

12^2 = 9 \cdot SR \implies 144 = 9 \cdot SR \implies SR = 16

cm

The altitude to the hypotenuse divides the original right triangle into two smaller similar right triangles.

3

Sum the segment lengths

PS

and

SR

to find the total length of hypotenuse

PR

PR = PS + SR = 9 + 16 = 25

cm

Point

S

lies directly on segment

PR

between endpoints

P

and

R

.

Anahtar Kavram

Geometric Mean Theorem and Pythagorean Theorem in Right Triangles

Tahmini Süre:1m 30s

Soru 182Soru →

A mountain zipline course consists of two connected ascending sections. The first section starts at base station $A$ and rises to intermediate platform $B$ at an angle of $30^\circ$ relative to the horizontal ground, covering a horizontal distance of $40\sqrt{3}$ meters. The second section rises from platform $B$ to peak platform $C$ at an angle of $45^\circ$ relative to the horizontal, covering a horizontal distance of $30$ meters. What is the total vertical height, in meters, of peak platform $C$ above base station $A$ ?

40

70

150

20\sqrt{3} + 30

40\sqrt{3} + 30

Cevabı ve açıklamayı göster

Cevap: $70$

Cevap

The total vertical height of peak platform C above base station A is 70 meters.

To find the total vertical height of peak platform C above base station A, determine the vertical rise of each section individually using special right triangle rules and sum them together. For the first section, the

30^\circ

incline forms a

30^\circ

-

60^\circ

-

90^\circ

right triangle where the horizontal distance of

40\sqrt{3}

meters is adjacent to the

30^\circ

angle. The vertical rise is opposite the

30^\circ

angle, so dividing

40\sqrt{3}

by

\sqrt{3}

gives a vertical rise of

40

meters. For the second section, the

45^\circ

incline forms a

45^\circ

-

45^\circ

-

90^\circ

right triangle where the vertical leg equals the horizontal leg, giving a vertical rise of

30

meters. Adding these two vertical heights yields

40 + 30 = 70

meters.

Adım Adım Çözüm

1

Calculate the vertical rise of the first section using

30^\circ

-

60^\circ

-

90^\circ

special right triangle properties.

The vertical height of the first section is

40

meters.

In a

30^\circ

-

60^\circ

-

90^\circ

triangle, the ratio of the side opposite the

30^\circ

angle (vertical rise) to the side opposite the

60^\circ

angle (horizontal distance) is

1 : \sqrt{3}

. Dividing the horizontal distance

40\sqrt{3}

by

\sqrt{3}

yields

40

meters.

2

Calculate the vertical rise of the second section using

45^\circ

-

45^\circ

-

90^\circ

special right triangle properties.

The vertical height of the second section is

30

meters.

In an isosceles right triangle (

45^\circ

-

45^\circ

-

90^\circ

), the two legs are equal in length. Since the horizontal distance is

30

meters, the vertical rise is also

30

meters.

3

Sum the vertical rises from both sections to find the total vertical height.

Total height =

40 + 30 = 70

meters.

The total vertical elevation of point C above point A is the sum of the vertical changes along each segment of the path.

Anahtar Kavram

Special Right Triangle Side Ratios (

30^\circ

-

60^\circ

-

90^\circ

and

45^\circ

-

45^\circ

-

90^\circ

)

Tahmini Süre:1m 30s

Soru 183Soru →

In a circle with center $O$ , a sector has a central angle measuring $\frac{2\pi}{5}$ radians and an arc length of $6\pi$ centimeters. What is the area, in square centimeters, of the sector?

15\pi

\frac{9\pi}{5}

45\pi

90\pi

225\pi

Cevabı ve açıklamayı göster

Cevap: $45\pi$

Cevap

The area of the sector is

45\pi

square centimeters.

To find the area of the sector, first determine the radius

r

of the circle using the arc length formula for radians,

s = r\theta

. Substituting

s = 6\pi

and

\theta = \frac{2\pi}{5}

yields

6\pi = r \left(\frac{2\pi}{5}\right)

, which simplifies to

r = 15

centimeters. Then, apply the sector area formula

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

. Substituting

r = 15

and

\theta = \frac{2\pi}{5}

gives

A = \frac{1}{2} (15^2) \left(\frac{2\pi}{5}\right) = 45\pi

square centimeters. Alternatively, using

A = \frac{1}{2} r s

directly gives

A = \frac{1}{2} (15)(6\pi) = 45\pi

square centimeters.

Adım Adım Çözüm

1

Find the radius of the circle using the arc length formula in radians.

r = 15

centimeters

The arc length formula for an angle in radians is

s = r\theta

. Substituting

s = 6\pi

and

\theta = \frac{2\pi}{5}

gives

6\pi = r \left(\frac{2\pi}{5}\right)

, which yields

r = 6\pi \cdot \frac{5}{2\pi} = 15

cm.

2

Calculate the area of the sector using the sector area formula.

A = 45\pi

square centimeters

The area of a sector with central angle

\theta

in radians is

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

. Substituting

r = 15

and

\theta = \frac{2\pi}{5}

gives

A = \frac{1}{2} (15^2) \left(\frac{2\pi}{5}\right) = \frac{1}{2} (225) \left(\frac{2\pi}{5}\right) = 45\pi

square centimeters.

Anahtar Kavram

Arc Length and Sector Area in Radians

Tahmini Süre:1m 0s

Soru 184Soru →

A drone begins at point $P$ and flies directly north for $24$ meters to point $Q$ . It then turns directly east and flies for $10$ meters to point $R$ . From point $R$ , the drone flies along a straight path directed $45^\circ$ south of east until it reaches point $S$ , which lies directly east of starting point $P$ . What is the distance, in meters, from point $P$ to point $S$ ?

Cevabı ve açıklamayı göster

Cevap: 34

Cevap

The total distance from point P to point S is 34 meters.

Flying north 24 meters places point R at a height of 24 meters above the horizontal line passing east through P. Returning to this horizontal line at point S along a path 45° south of east forms a 45°-45°-90° right triangle with a vertical leg of 24 meters. Because the legs of a 45°-45°-90° right triangle are congruent, the horizontal leg is also 24 meters. Combining this with the initial 10 meters of eastward travel gives a total distance from P to S of 10 + 24 = 34 meters.

Adım Adım Çözüm

1

Determine the vertical and horizontal position of point R relative to point P.

Point R is located 24 meters north and 10 meters east of point P.

The flight 24 meters north sets the vertical distance to 24 meters, and the flight 10 meters east sets the horizontal displacement to 10 meters.

2

Apply the properties of a 45°-45°-90° special right triangle to find the horizontal distance from point R to point S.

The horizontal distance traveled between point R and point S is 24 meters.

Since point S lies directly east of point P (at vertical height 0), the vertical drop from point R to point S is 24 meters. A line angled 45° south of east creates a 45°-45°-90° right triangle whose two leg lengths are equal, so the horizontal leg length equals the vertical leg length of 24 meters.

3

Calculate the total horizontal distance from point P to point S.

34 meters

Sum the initial east displacement of 10 meters from P to R with the additional horizontal displacement of 24 meters from R to S: 10 + 24 = 34 meters.

Anahtar Kavram

45°-45°-90° Special Right Triangle Properties and Planar Displacement

Tahmini Süre:1m 30s

Soru 185Soru →

Quadrilateral $ABCD$ has vertices $A(-2, 1)$ , $B(2, 4)$ , $C(5, 0)$ , and $D(1, -3)$ in the standard $(x,y)$ coordinate plane. What is the area, in square units, of quadrilateral $ABCD$ ?

12.5

15

20

25

50

Cevabı ve açıklamayı göster

Cevap: $25$

Cevap

25

Using the distance formula

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

, each side length of quadrilateral

ABCD

is calculated to be

5

units. Evaluating the slopes of adjacent sides shows that the slope of

AB

is

\frac{3}{4}

and the slope of

BC

is

-\frac{4}{3}

. Since their product is

-1

, adjacent sides are perpendicular. A quadrilateral with four equal sides and right angles is a square, and its area is

5^2 = 25

square units.

Adım Adım Çözüm

1

Calculate the length of side

AB

using the distance formula.

AB = \sqrt{(2 - (-2))^2 + (4 - 1)^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5

.

The distance formula finds the exact side length of the quadrilateral.

2

Calculate adjacent side length

BC

and check the slopes to determine the figure type.

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

. The slope of

AB

is

\frac{3}{4}

and the slope of

BC

is

-\frac{4}{3}

.

Because all sides are equal to

5

and adjacent slopes are negative reciprocals, the figure is a square.

3

Calculate the area of the square.

\text{Area} = \text{side}^2 = 5^2 = 25

.

The area of a square with side length

s

is

s^2

.

Anahtar Kavram

Finding the area of a quadrilateral in the coordinate plane by verifying side lengths and right angles using distance and slope formulas.

Soru 186Soru →

In parallelogram $ABCD$ , diagonals $AC$ and $BD$ intersect at point $E$ . In the standard $(x,y)$ coordinate plane, vertex $A$ is located at $(-3, 2)$ and point $E$ is located at $(2, 1)$ . What are the coordinates of vertex $C$ ?

(-0.5, 1.5)

(3, -5)

(5, -1)

(7, 0)

(7, 4)

Cevabı ve açıklamayı göster

Cevap: $(7, 0)$

Cevap

The coordinates of vertex

C

are

(7, 0)

.

A fundamental property of any parallelogram is that its diagonals bisect each other. Therefore, the intersection point

E(2, 1)

must be the midpoint of diagonal

AC

. Using the midpoint formula,

\frac{-3 + x_C}{2} = 2

gives

x_C = 7

, and

\frac{2 + y_C}{2} = 1

gives

y_C = 0

. Thus, the coordinates of vertex

C

are

(7, 0)

.

Adım Adım Çözüm

1

Identify the key geometric property of parallelograms.

The diagonals of a parallelogram bisect each other, which means point

E

is the midpoint of diagonal

AC

.

By definition of diagonal bisection in any parallelogram, the intersection point of the diagonals is the midpoint of both diagonal segments.

2

Set up the midpoint formula for segment

AC

with midpoint

E(2, 1)

.

(\frac{x_A + x_C}{2}, \frac{y_A + y_C}{2}) = (2, 1)

, where

x_A = -3

and

y_A = 2

.

The midpoint coordinates are the averages of the endpoint coordinates.

3

Solve for the

x

-coordinate of vertex

C

.

\frac{-3 + x_C}{2} = 2 \implies -3 + x_C = 4 \implies x_C = 7

.

Multiply by 2 and add 3 to isolate

x_C

.

4

Solve for the

y

-coordinate of vertex

C

.

\frac{2 + y_C}{2} = 1 \implies 2 + y_C = 2 \implies y_C = 0

.

Multiply by 2 and subtract 2 to isolate

y_C

.

Anahtar Kavram

Diagonals of a Parallelogram Bisect Each Other

Alternatif Yöntem

Use vector translations: The vector from

A(-3, 2)

to

E(2, 1)

is

\langle 2 - (-3), 1 - 2 \rangle = \langle 5, -1 \rangle

. Since

E

is the midpoint of

AC

, the vector from

E

to

C

is identical. Adding

\langle 5, -1 \rangle

to

E(2, 1)

gives

C(2 + 5, 1 - 1) = (7, 0)

.

Tahmini Süre:1m 15s

Soru 187Soru →

A decorative circular stained-glass window has a radius of $10$ inches. A specific colored section of the window forms a sector with a central angle measuring $108^\circ$ . What is the total perimeter, in inches, of this stained-glass sector?

6\pi

3\pi + 20

6\pi + 10

6\pi + 20

30\pi + 20

Cevabı ve açıklamayı göster

Cevap: $6\pi + 20$

Cevap

The total perimeter of the sector is

6\pi + 20

inches.

The arc length for a central angle of

108^\circ

in a circle of radius

10

is

\frac{108}{360} \times 2\pi(10) = 6\pi

inches. Adding the two bounding radii (

10 + 10 = 20

) yields the full sector perimeter of

6\pi + 20

inches.

Adım Adım Çözüm

1

Calculate the arc length of the sector

Arc length =

\frac{108^\circ}{360^\circ} \cdot 2\pi(10) = \frac{3}{10} \cdot 20\pi = 6\pi

inches.

The curved boundary of a sector is a fraction of the circle's full circumference defined by the central angle ratio.

2

Calculate the total perimeter of the sector

Perimeter = Arc length +

2 \cdot \text{radius} = 6\pi + 2(10) = 6\pi + 20

inches.

A sector's perimeter consists of its outer curved arc plus its two straight radial edges.

Anahtar Kavram

Perimeter of a Sector

Tahmini Süre:1m 0s

Soru 188Soru →

In the figure below, quadrilateral $ABCD$ is composed of two adjacent right triangles, $\triangle ABD$ and $\triangle BCD$ , sharing side $BD$ . In $\triangle ABD$ , the right angle is at vertex $A$ , and the measure of $\angle ABD$ is $60^\circ$ . In $\triangle BCD$ , the right angle is at vertex $C$ , and the measure of $\angle BDC$ is $45^\circ$ . If the length of segment $AB$ is $6$ inches, what is the length, in inches, of segment $BC$ ?

6\sqrt{2}

6\sqrt{3}

12

12\sqrt{2}

4\sqrt{3}

Cevabı ve açıklamayı göster

Cevap: $6\sqrt{2}$

Cevap

The length of segment

BC

is

6\sqrt{2}

inches.

In the

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

triangle

ABD

, short leg

AB = 6

gives hypotenuse

BD = 12

. In the

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

triangle

BCD

, hypotenuse

BD = 12

gives leg

BC = \frac{12}{\sqrt{2}} = 6\sqrt{2}

inches.

Adım Adım Çözüm

1

Analyze

\triangle ABD

using

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

special right triangle ratios.

Since

\angle A = 90^\circ

and

\angle ABD = 60^\circ

,

\angle ADB = 30^\circ

. The side opposite

30^\circ

is

AB = 6

. Therefore, the hypotenuse

BD = 2 \times AB = 2(6) = 12

.

In a

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

triangle, the hypotenuse is twice the shorter leg.

2

Analyze

\triangle BCD

using

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

special right triangle ratios.

Triangle

BCD

is an isosceles right triangle with right angle at

C

and hypotenuse

BD = 12

. The legs are equal, so

BC = CD

.

In a

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

triangle, hypotenuse =

\text{leg} \times \sqrt{2}

.

3

Solve for leg

BC

and rationalize the denominator.

BC = \frac{BD}{\sqrt{2}} = \frac{12}{\sqrt{2}} = \frac{12\sqrt{2}}{2} = 6\sqrt{2}

.

Dividing the hypotenuse by

\sqrt{2}

yields the leg length in standard simplified radical form.

Anahtar Kavram

Applying

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

and

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

special right triangle side ratio rules across multi-step figures.

Tahmini Süre:1m 15s

Soru 189Soru →

In any trapezoid, the length of the midsegment connecting the midpoints of the non-parallel legs is equal to half the difference of the lengths of the two parallel bases.

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: False

Cevap

False. The length of the midsegment of a trapezoid is equal to half the sum of the lengths of the parallel bases, not half the difference.

The statement is false because the Trapezoid Midsegment Theorem dictates that the length of the midsegment connecting the midpoints of the non-parallel legs is equal to half the sum (the average) of the lengths of the parallel bases,

\frac{b_1 + b_2}{2}

. Half the difference of the base lengths gives the length of the segment connecting the midpoints of the diagonals.

Adım Adım Çözüm

1

Recall the definition of a trapezoid midsegment.

The midsegment of a trapezoid is the line segment joining the midpoints of its non-parallel legs.

Identifying the geometric figure and segment being described is necessary to apply the correct theorem.

2

Apply the Trapezoid Midsegment Theorem.

The length

m

of the midsegment for a trapezoid with parallel bases

b_1

and

b_2

is given by

m = \frac{b_1 + b_2}{2}

.

The theorem states that the midsegment length is the average (half the sum) of the two base lengths.

3

Compare the theorem formula to the given statement.

The statement claims the length is

\frac{|b_1 - b_2|}{2}

(half the difference), which contradicts the true formula

\frac{b_1 + b_2}{2}

. Therefore, the statement is false.

Evaluating the mathematical validity of the statement determines the correct answer.

Anahtar Kavram

Trapezoid Midsegment Theorem

Soru 190Soru →

A robotic arm mounted on a flat work surface pivots through a central angle of $\theta$ degrees, sweeping out a sector of a circle with a radius of $15$ centimeters. If the total perimeter of the swept sector is $(30 + 5\pi)$ centimeters, what is the value of $\theta$ ?

8^\circ

30^\circ

45^\circ

60^\circ

120^\circ

Cevabı ve açıklamayı göster

Cevap: $60^\circ$

Cevap

The measure of the central angle

\theta

is

60^\circ

.

The perimeter of a sector consists of two straight radii of length

r

plus the arc length

s

, so

\text{Perimeter} = 2r + s

. Given

r = 15

cm and a perimeter of

(30 + 5\pi)

cm, we find

2(15) + s = 30 + 5\pi

, which means

s = 5\pi

cm. Using the arc length formula

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

, we set

5\pi = \frac{\theta}{360^\circ} \times 30\pi

. Simplifying yields

\frac{\theta}{12^\circ} = 5

, giving

\theta = 60^\circ

.

Adım Adım Çözüm

1

Express the perimeter of a sector in terms of radius

r

and arc length

s

.

\text{Perimeter} = 2r + s

A sector's boundary consists of two straight radii and one curved arc.

2

Substitute

r = 15

cm and the given perimeter

(30 + 5\pi)

cm to solve for the arc length

s

.

30 + s = 30 + 5\pi \implies s = 5\pi \text{ cm}

Subtracting the combined length of the two straight edges (

2 \times 15 = 30

) isolates the arc length.

3

Set up the arc length formula using degrees:

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

.

5\pi = \frac{\theta}{360^\circ} \times 2\pi(15) = \frac{30\pi \theta}{360^\circ} = \frac{\pi \theta}{12^\circ}

The arc length represents the fraction

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

of the circle's total circumference

2\pi r

.

4

Solve the equation for

\theta

.

5\pi = \frac{\pi \theta}{12^\circ} \implies 5 = \frac{\theta}{12^\circ} \implies \theta = 60^\circ

Dividing both sides by

\pi

and multiplying by

12^\circ

isolates

\theta

.

Anahtar Kavram

Perimeter of a Sector and Arc Length

Tahmini Süre:1m 15s

Soru 191Soru →

A regular hexagon $ABCDEF$ has a perpendicular distance of $12\sqrt{3}$ inches between its two parallel opposite sides. What is the perimeter, in inches, of hexagon $ABCDEF$ ?

Cevabı ve açıklamayı göster

Cevap: 72

Cevap

The perimeter of the regular hexagon is 72 inches.

A regular hexagon with side length

s

can be partitioned from its center into 6 congruent equilateral triangles of side length

s

. Dropping an altitude from the center to any side creates two

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

right triangles. In each right triangle, the side opposite the

60^\circ

angle (the altitude) has length

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

. The total perpendicular distance between two parallel opposite sides of the hexagon equals twice this altitude,

s\sqrt{3}

. Setting

s\sqrt{3} = 12\sqrt{3}

gives

s = 12

inches. The perimeter of the regular hexagon is

6 \times 12 = 72

inches.

Adım Adım Çözüm

1

Express the perpendicular distance between parallel opposite sides of a regular hexagon in terms of its side length

s

.

The distance between opposite sides is

s\sqrt{3}

.

A regular hexagon with side length

s

consists of 6 congruent equilateral triangles. The altitude of each equilateral triangle divides it into two

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

right triangles with legs

s/2

and

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

. The distance between opposite parallel sides spans two altitudes, which equals

2 \times \frac{s\sqrt{3}}{2} = s\sqrt{3}

.

2

Solve for the side length

s

.

s = 12

inches.

Equating the given distance

12\sqrt{3}

to

s\sqrt{3}

yields

s = 12

.

3

Calculate the total perimeter of the hexagon.

Perimeter

= 72

inches.

A regular hexagon has 6 equal sides, so its perimeter is

6 \times 12 = 72

inches.

Anahtar Kavram

Applying

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

special right triangle relationships to regular polygons

Soru 192Soru →

A garden bed is designed in the shape of a sector of a circle with a central angle measuring $120^\circ$ . If the total perimeter of the sector-shaped garden bed is $24 + 8\pi$ feet, what is the area of the garden bed, in square feet?

8\pi

24\pi

48\pi

96\pi

144\pi

Cevabı ve açıklamayı göster

Cevap: $48\pi$

Cevap

The area of the garden bed is

48\pi

square feet.

The perimeter of a sector is given by

2r + \frac{\theta}{360^\circ}(2\pi r)

. Substituting

\theta = 120^\circ

gives

2r + \frac{2\pi r}{3} = 24 + 8\pi

, which simplifies to

r = 12

feet. Using the sector area formula

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

with

r = 12

yields

\frac{1}{3}\pi(144) = 48\pi

square feet.

Adım Adım Çözüm

1

Set up the formula for the perimeter of a sector.

Perimeter

= 2r + \text{Arc Length} = 2r + \left(\frac{\theta}{360^\circ}\right)(2\pi r)

A sector's perimeter consists of two straight radii and the curved arc boundary.

2

Substitute

\theta = 120^\circ

into the perimeter expression and solve for the radius

r

.

2r + \left(\frac{120^\circ}{360^\circ}\right)(2\pi r) = 24 + 8\pi \implies 2r + \frac{2\pi r}{3} = 24 + 8\pi \implies r\left(2 + \frac{2\pi}{3}\right) = 12\left(2 + \frac{2\pi}{3}\right) \implies r = 12

feet

Equating the algebraic expression to the given perimeter allows finding the circle radius.

3

Calculate the sector area using

r = 12

feet and

\theta = 120^\circ

.

Area

= \left(\frac{120^\circ}{360^\circ}\right) \pi r^2 = \frac{1}{3} \pi (12)^2 = \frac{144\pi}{3} = 48\pi

square feet

The area of a sector is the fraction of the total circle's area defined by the ratio of central angle to

360^\circ

.

Anahtar Kavram

Perimeter and Area of a Circle Sector

Tahmini Süre:1m 0s

Soru 193Soru →

A vertical flagpole stands perpendicular to level ground. Two support cables are attached from the top of the flagpole to ground anchors located on opposite sides of the pole. The first cable makes a $60^\circ$ angle with the ground and its ground anchor is $10\sqrt{3}$ feet from the base of the pole. The second cable makes a $45^\circ$ angle with the ground. What is the sum of the lengths, in feet, of the two support cables?

20\sqrt{3} + 30\sqrt{2}

10\sqrt{3} + 30\sqrt{2}

30\sqrt{3} + 30\sqrt{2}

20\sqrt{3} + 30

50\sqrt{5}

Cevabı ve açıklamayı göster

Cevap: $20\sqrt{3} + 30\sqrt{2}$

Cevap

20\sqrt{3} + 30\sqrt{2}

feet

The first support cable forms a 30°-60°-90° right triangle with the flagpole and ground. Given the adjacent leg to the 60° angle is

10\sqrt{3}

feet, the opposite leg (the flagpole's height) is

10\sqrt{3} \times \sqrt{3} = 30

feet, and the hypotenuse (first cable) is

2 \times 10\sqrt{3} = 20\sqrt{3}

feet. The second cable forms a 45°-45°-90° right triangle sharing the 30-foot height as one leg. Thus, the hypotenuse (second cable) is

30\sqrt{2}

feet. Adding both cable lengths gives

20\sqrt{3} + 30\sqrt{2}

feet.

Adım Adım Çözüm

1

Analyze the first right triangle formed by the pole and the first cable

The triangle is a

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

triangle with the ground angle equal to

60^\circ

. The side adjacent to

60^\circ

(shorter leg) is

10\sqrt{3}

ft.

The cable makes a

60^\circ

angle with the horizontal ground, making the angle at the top of the pole

30^\circ

.

2

Calculate the height of the flagpole and the length of the first cable

Flagpole height

= 10\sqrt{3} \cdot \sqrt{3} = 30

ft. First cable length (hypotenuse)

= 2 \cdot 10\sqrt{3} = 20\sqrt{3}

ft.

In a

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

triangle with shorter leg

x

, the longer leg is

x\sqrt{3}

and the hypotenuse is

2x

.

3

Calculate the length of the second cable using the second right triangle

Second cable length (hypotenuse)

= 30\sqrt{2}

ft.

The second triangle is a

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

right triangle with leg equal to the pole height (

30

ft). The hypotenuse is

\text{leg} \cdot \sqrt{2}

.

4

Add the lengths of the two support cables

Total length

= 20\sqrt{3} + 30\sqrt{2}

ft.

Summing the two hypotenuse lengths gives the total combined cable length.

Anahtar Kavram

Properties of

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

and

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

special right triangles

Tahmini Süre:1m 30s

Soru 194Soru →

In rectangle $ABCD$ , diagonals $AC$ and $BD$ intersect at point $E$ . If the measure of $\angle AEB$ is $120^\circ$ and $AC = 16$ , what is the length of side $BC$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The length of side

BC

is

8

.

In any rectangle, the diagonals are congruent and bisect each other. Given

AC = 16

, the distance from the intersection point

E

to any vertex is

8

, so

BE = EC = 8

. Because

\angle AEB

and

\angle BEC

lie along the straight diagonal line

AC

, they are supplementary, giving

\angle BEC = 180^\circ - 120^\circ = 60^\circ

. Triangle

BEC

is an isosceles triangle with

BE = EC = 8

and a vertex angle of

60^\circ

, which forces it to be equilateral. Consequently, all sides of

\triangle BEC

are equal, so

BC = 8

.

Adım Adım Çözüm

1

Calculate the lengths of the diagonal segments from the center point

E

.

BE = EC = 8

The diagonals of a rectangle are congruent and bisect each other, so each half-diagonal equals half of

AC

.

2

Determine the measure of adjacent angle

\angle BEC

.

\angle BEC = 60^\circ

Angles

\angle AEB

and

\angle BEC

form a straight line (linear pair), so their sum is

180^\circ

.

3

Analyze

\triangle BEC

to find the length of side

BC

.

BC = 8

An isosceles triangle with a

60^\circ

vertex angle has base angles of

60^\circ

as well, making it an equilateral triangle where all sides are equal to

8

.

Anahtar Kavram

Diagonals of a rectangle are equal in length and bisect each other, dividing the rectangle into two pairs of congruent isosceles triangles.

Tahmini Süre:1m 15s

Soru 195Soru →

A rhombus has a perimeter of 40 centimeters and one diagonal of length 12 centimeters. What is the area, in square centimeters, of the rhombus?

48

60

120

192

96

Cevabı ve açıklamayı göster

Cevap: 96

Cevap

96 square centimeters

Because all four sides of a rhombus are congruent, a perimeter of 40 centimeters gives a side length of 10 centimeters. The diagonals of a rhombus intersect at right angles and bisect each other. Half of the given diagonal is 6 centimeters. By applying the Pythagorean theorem (

6^2 + b^2 = 10^2

), the second half-diagonal is found to be 8 centimeters, which means the full length of the second diagonal is 16 centimeters. Using the rhombus area formula

\text{Area} = \frac{1}{2} \cdot d_1 \cdot d_2

, the area is

\frac{1}{2} \cdot 12 \cdot 16 = 96

square centimeters.

Adım Adım Çözüm

1

Calculate the side length of the rhombus from its perimeter.

Side length

s = \frac{40}{4} = 10\text{ cm}

.

All four sides of a rhombus are equal in length.

2

Use the properties of rhombus diagonals to find the length of the second diagonal.

Half of the given diagonal is

\frac{12}{2} = 6\text{ cm}

. In the right triangle formed by the half-diagonals and a side:

6^2 + b^2 = 10^2 \implies 36 + b^2 = 100 \implies b = 8\text{ cm}

. Thus, the second diagonal

d_2 = 2 \times 8 = 16\text{ cm}

.

The diagonals of a rhombus are perpendicular bisectors of each other.

3

Calculate the area of the rhombus using the diagonal formula.

\text{Area} = \frac{1}{2} \times d_1 \times d_2 = \frac{1}{2} \times 12 \times 16 = 96\text{ sq cm}.

The area of any rhombus is equal to half the product of the lengths of its diagonals.

Anahtar Kavram

Properties of Rhombus Diagonals and Area Calculation

Soru 196Soru →

A circular stage turntable has a radius of $9$ meters. A wedge-shaped performance zone on the turntable is formed by a sector with a central angle measuring $100^\circ$ . A safety border is installed along the entire boundary of this performance zone, which consists of its curved arc and both straight radius edges. What is the total length, in meters, of the safety border?

5\pi

5\pi + 9

5\pi + 18

2.5\pi + 18

22.5\pi

Cevabı ve açıklamayı göster

Cevap: $5\pi + 18$

Cevap

The total length of the safety border is

5\pi + 18

meters.

To find the total length of the border surrounding the sector, calculate the arc length of the sector using

s = \frac{\theta}{360^\circ} \times 2\pi r = \frac{100}{360} \times 18\pi = 5\pi

meters. Then, add the two straight radial sides (

2r = 2 \times 9 = 18

meters) to get the complete perimeter of

5\pi + 18

meters.

Adım Adım Çözüm

1

Calculate the arc length (

s

) of the circular sector.

s = \frac{100^\circ}{360^\circ} \times 2\pi(9) = \frac{5}{18} \times 18\pi = 5\pi

meters.

The arc length formula for a sector measured in degrees is

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

.

2

Calculate the total perimeter (

P

) of the sector.

P = s + 2r = 5\pi + 2(9) = 5\pi + 18

meters.

The boundary of a sector includes the curved arc length plus two straight radial sides.

Anahtar Kavram

Perimeter of a Circle Sector

Tahmini Süre:1m 0s

Soru 197Soru →

The tip of a pendulum swings through a central angle measuring $\frac{\pi}{3}$ radians, tracing an arc length of $4\pi$ centimeters. What is the area, in square centimeters, of the sector swept out by the pendulum?

12\pi

24\pi

36\pi

48\pi

144\pi

Cevabı ve açıklamayı göster

Cevap: $24\pi$

Cevap

24\pi

square centimeters

To find the area of the sector, first determine the radius of the pendulum's circular path using the arc length formula

s = r\theta

. Substituting

s = 4\pi

and

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

gives

r = 12

cm. Then, substitute the radius and central angle into the sector area formula

A = \frac{1}{2}r^2\theta = \frac{1}{2}(12)^2\left(\frac{\pi}{3}\right) = 24\pi

square centimeters.

Adım Adım Çözüm

1

Calculate the radius of the circle using the arc length formula in radians.

Using

s = r\theta

with

s = 4\pi

and

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

gives

4\pi = r\left(\frac{\pi}{3}\right) \implies r = 12

cm.

The radius is required to calculate the sector area.

2

Calculate the area of the sector using the radian sector area formula.

A = \frac{1}{2}r^2\theta = \frac{1}{2}(12)^2\left(\frac{\pi}{3}\right) = \frac{1}{2}(144)\left(\frac{\pi}{3}\right) = 24\pi

square centimeters.

Multiplying half the squared radius by the radian measure of the central angle yields the sector area.

Anahtar Kavram

Arc Length and Sector Area in Radians

Tahmini Süre:1m 0s

Soru 198Soru →

A circular section of a community park is bounded by two straight footpaths meeting at the park's center at a $75^\circ$ angle, and an outer curved walking trail. If the length of the outer curved walking trail is $5\pi$ meters, what is the area, in square meters, of this section of the park?

25\pi

30\pi

60\pi

120\pi

144\pi

Cevabı ve açıklamayı göster

Cevap: $30\pi$

Cevap

The area of this section of the park is

30\pi

square meters.

The central angle of

75^\circ

represents

\frac{75}{360} = \frac{5}{24}

of the circle. Setting the arc length

5\pi = \frac{5}{24} (2\pi r)

gives

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

, which simplifies to

r = 12

meters. The sector area is

\frac{5}{24} \cdot \pi (12)^2 = 30\pi

square meters.

Adım Adım Çözüm

1

Find the radius of the circular park section using the arc length formula.

r = 12

meters

The arc length formula is

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

. Substituting

s = 5\pi

and

\theta = 75^\circ

gives

5\pi = \frac{75}{360} \cdot 2\pi r \implies 5\pi = \frac{5}{24} \cdot 2\pi r \implies 5\pi = \frac{5\pi r}{12} \implies r = 12

meters.

2

Calculate the sector area using the radius and central angle.

Area =

30\pi

square meters

The sector area formula is

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

. Substituting

\theta = 75^\circ

and

r = 12

gives

A = \frac{75}{360} \cdot \pi (12)^2 = \frac{5}{24} \cdot 144\pi = 30\pi

square meters.

Anahtar Kavram

Determining Sector Area from Central Angle and Arc Length

Soru 199Soru →

A rotary lawn sprinkler sweeps out a sector-shaped region of a yard with a central angle of $150^\circ$ . If the area of the irrigated sector is $60\pi$ square feet, what is the perimeter, in feet, of the irrigated lawn sector?

10\pi

10\pi + 12

10\pi + 24

5\pi + 24

20\pi + 24

Cevabı ve açıklamayı göster

Cevap: $10\pi + 24$

Cevap

The perimeter of the irrigated lawn sector is

10\pi + 24

feet.

The area of a sector with central angle

150^\circ

is

\frac{150^\circ}{360^\circ} \times \pi r^2 = \frac{5}{12}\pi r^2

. Setting this equal to

60\pi

yields

r^2 = 144

, so the radius

r = 12

feet. The arc length is

\frac{5}{12} \times 2\pi(12) = 10\pi

feet. The perimeter of a sector is the arc length plus two radii (

2r

), giving

10\pi + 2(12) = 10\pi + 24

feet.

Adım Adım Çözüm

1

Find the radius of the circle using the sector area formula.

r = 12

feet

The formula for sector area is

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

. Substituting

\theta = 150^\circ

and

\text{Area} = 60\pi

gives

\frac{150}{360} \cdot \pi r^2 = 60\pi

, which simplifies to

\frac{5}{12} r^2 = 60

, so

r^2 = 144

and

r = 12

.

2

Calculate the arc length of the sector.

Arc length

= 10\pi

feet

The formula for arc length is

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

. Substituting

\theta = 150^\circ

and

r = 12

yields

s = \frac{5}{12} \cdot 2\pi (12) = 10\pi

.

3

Calculate the total perimeter of the sector.

Perimeter

= 10\pi + 24

feet

The perimeter of a sector consists of the curved arc length plus two straight radial edges:

\text{Perimeter} = s + 2r = 10\pi + 2(12) = 10\pi + 24

.

Anahtar Kavram

Perimeter of a Circle Sector

Tahmini Süre:1m 30s

Soru 200Soru →

A circular stained-glass window panel has a radius of $15$ centimeters. A specific red sector within the panel has an outer arc length of $6\pi$ centimeters. What is the area, in square centimeters, of this red sector?

15\pi

30\pi

45\pi

90\pi

225\pi

Cevabı ve açıklamayı göster

Cevap: $45\pi$

Cevap

45\pi

square centimeters

The total circumference of the circle is

2\pi(15) = 30\pi

cm. The sector's arc length of

6\pi

cm is

\frac{6\pi}{30\pi} = \frac{1}{5}

of the entire circle. Since the sector area is proportional to the arc length, the area of the sector is

\frac{1}{5}

of the total area. The total area is

\pi(15)^2 = 225\pi

sq cm, so the sector area is

\frac{1}{5}(225\pi) = 45\pi

sq cm. Alternatively, using the sector area formula

A = \frac{1}{2} r s

, we get

A = \frac{1}{2}(15)(6\pi) = 45\pi

sq cm.

Adım Adım Çözüm

1

Calculate the total circumference of the circle.

Circumference

C = 2\pi r = 2\pi(15) = 30\pi

cm

Knowing the total circumference allows us to determine what fraction of the circle the arc represents.

2

Determine the fractional portion of the circle corresponding to the arc.

Fraction

= \frac{\text{Arc Length}}{\text{Circumference}} = \frac{6\pi}{30\pi} = \frac{1}{5}

The central angle ratio is equivalent to the ratio of arc length to total circumference.

3

Calculate the total area of the circle.

Total Area

A_{\text{total}} = \pi r^2 = \pi (15)^2 = 225\pi

cm

^2

The sector area will be the same fraction of the total area as the arc length is of the circumference.

4

Multiply the fraction by the total area to find the sector area.

Sector Area

= \frac{1}{5} \times 225\pi = 45\pi

cm

^2

Applying the fraction

\frac{1}{5}

to the full circle area yields the targeted sector area.

Anahtar Kavram

Relationship between Arc Length and Sector Area in Circle Geometry

Plane Geometry

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