Triangles and Pythagorean Theorem Alıştırma Soruları — Armed Services Vocational Aptitude Battery (ASVAB)

Determine the missing side length based on the geometric properties of a right triangle.

Aşağıdaki boşlukları doldurun

In a right triangle, the two shorter sides (legs) measure

9

cm and

12

cm. According to the Pythagorean theorem, the length of the hypotenuse is cm.

Cevabı ve açıklamayı göster

Cevap

15

By applying the Pythagorean theorem (

a^2 + b^2 = c^2

) to a right triangle with legs of

9

and

12

, we square both numbers to get

81

and

144

. Their sum is

225

. The hypotenuse is the square root of

225

, which is

15

.

Adım Adım Çözüm

1

Identify the known sides of the right triangle.

The legs (let's call them

a

and

b

) measure

9

cm and

12

cm.

The Pythagorean theorem requires knowing the lengths of two sides to find the third.

2

Set up the Pythagorean theorem equation.

a^2 + b^2 = c^2

, which becomes

9^2 + 12^2 = c^2

.

This formula relates the squares of the legs to the square of the hypotenuse (

c

).

3

Square the lengths of the two legs and add them together.

81 + 144 = 225

This calculation determines the value of

c^2

.

4

Take the square root of the sum to find the hypotenuse.

\sqrt{225} = 15

Since

c^2 = 225

, taking the square root gives the actual length of the hypotenuse (

c

).

Anahtar Kavram

Pythagorean Theorem

Soru 2Soru →

A stained glass artist cuts a piece of glass into the shape of a right triangle. The two perpendicular legs of the triangle measure $30$ centimeters and $40$ centimeters. What is the total perimeter of this glass triangle in centimeters?

Cevabı ve açıklamayı göster

Cevap: 120

Cevap

120

The correct answer is found by first determining the length of the hypotenuse using the Pythagorean theorem. Since the legs are

30

and

40

, the hypotenuse is

\sqrt{30^2 + 40^2} = \sqrt{2500} = 50

. The perimeter is then calculated by adding the lengths of all three sides:

30 + 40 + 50 = 120

.

Adım Adım Çözüm

1

Use the Pythagorean theorem to find the length of the missing hypotenuse.

c = \sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50

centimeters.

The perimeter requires all three side lengths, but only the two legs are provided.

2

Calculate the total perimeter of the right triangle.

Perimeter =

30 + 40 + 50 = 120

centimeters.

The perimeter is the total distance around the outside of the shape, found by summing all side lengths.

Anahtar Kavram

Applying the Pythagorean theorem to find a missing side length, then using all three side lengths to calculate the perimeter of a right triangle.

Soru 3Soru →

Determine the required dimensions for the construction project.

Aşağıdaki boşlukları doldurun

A construction crew is building a wooden ramp with a right triangular cross-section. The horizontal base of the ramp measures

24\text{ feet}

and it rises to a vertical height of

7\text{ feet}

. According to the Pythagorean Theorem, the length of the sloped surface is feet, and the total perimeter of the ramp's triangular profile is feet.

Cevabı ve açıklamayı göster

Cevap

The sloped surface is 25 feet, and the total perimeter is 56 feet.

The sloped surface corresponds to the hypotenuse of a right triangle with legs of

7

and

24

feet. Applying the Pythagorean Theorem,

7^2 + 24^2 = 49 + 576 = 625

. Taking the square root of

625

gives

25\text{ feet}

for the sloped surface. The total perimeter is the sum of the base, height, and sloped surface:

24 + 7 + 25 = 56\text{ feet}

.

Adım Adım Çözüm

1

Identify the known sides of the right triangle.

The legs of the right triangle are

a = 7

and

b = 24

.

The horizontal base and vertical height form the two legs that intersect at a right angle.

2

Apply the Pythagorean Theorem to find the length of the hypotenuse.

c^2 = 7^2 + 24^2 = 49 + 576 = 625

. Therefore,

c = \sqrt{625} = 25

.

The sloped surface represents the hypotenuse (

c

) of the right triangle.

3

Calculate the perimeter of the triangle.

7 + 24 + 25 = 56

.

The perimeter is found by summing the lengths of all three sides of the triangular cross-section.

Anahtar Kavram

Using the Pythagorean Theorem to find the hypotenuse of a right triangle, and applying it to calculate the perimeter.

Soru 4Soru →

Solve the following word problem by applying properties of right triangles.

Aşağıdaki boşlukları doldurun

A 25-foot ladder is leaning against a vertical wall, with its base placed 7 feet away from the bottom of the wall. If the top of the ladder slides exactly 4 feet straight down the wall, the base of the ladder will slide feet further away from the wall.

Cevabı ve açıklamayı göster

Cevap

8

By applying the Pythagorean theorem to the initial position, we establish that the ladder reached a height of 24 feet (

25^2 - 7^2 = 24^2

). When it slips 4 feet down, the new height becomes 20 feet. Using the Pythagorean theorem again for the new position with a constant hypotenuse of 25, the new base distance is 15 feet (

25^2 - 20^2 = 15^2

). The ladder started 7 feet away, so it moved an additional 8 feet (

15 - 7 = 8

).

Adım Adım Çözüm

1

Find the initial height of the top of the ladder on the wall using the Pythagorean theorem.

a^2 + 7^2 = 25^2

, which simplifies to

a^2 + 49 = 625

. Solving for

a^2

gives

576

, so

a = 24

feet.

The ladder, the wall, and the ground form a right triangle where the ladder is the hypotenuse.

2

Calculate the new height of the top of the ladder after it slides down the wall.

24 - 4 = 20

feet.

The top of the ladder slid exactly 4 feet downwards, reducing the vertical leg of our right triangle.

3

Find the new distance of the base of the ladder from the wall.

b^2 + 20^2 = 25^2

, which simplifies to

b^2 + 400 = 625

. Solving for

b^2

gives

225

, so

b = 15

feet.

The ladder's overall length does not change, creating a new right triangle with a height of 20 and a hypotenuse of 25.

4

Calculate the difference between the new base distance and the original base distance.

15 - 7 = 8

feet.

The question specifically asks how much further the base slid away from the wall, not the final total distance.

Anahtar Kavram

Applying the Pythagorean theorem iteratively to linked right triangles that share a common hypotenuse length

Soru 5Soru →

In quadrilateral $ABCD$ , $\angle B$ and $\angle D$ are right angles. The lengths of three sides are given as $AB = 7$ , $BC = 24$ , and $CD = 15$ . What is the length of side $AD$ ?

8

10

20

46

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The length of side

AD

is 20.

Drawing the diagonal

AC

creates two right triangles that share a hypotenuse. The first triangle (

ABC

) has legs of

7

and

24

, giving a hypotenuse of

25

(

7^2 + 24^2 = 25^2

). In the second triangle (

ADC

), this

25

serves as the hypotenuse while

CD

is a leg of

15

. Solving for the remaining leg gives

20

(

25^2 - 15^2 = 400

, and

\sqrt{400} = 20

).

Adım Adım Çözüm

1

Visualize or draw diagonal

AC

to split the quadrilateral into two right triangles.

Right

\triangle ABC

and right

\triangle ADC

are formed, sharing hypotenuse

AC

.

Since

\angle B

and

\angle D

are right angles, the segment connecting

A

and

C

must be the hypotenuse for both triangles.

2

Apply the Pythagorean theorem to

\triangle ABC

to find the length of

AC

.

AC^2 = 7^2 + 24^2 = 49 + 576 = 625

, which means

AC = 25

.

The lengths of both legs (

AB

and

BC

) are known, allowing calculation of the hypotenuse.

3

Apply the Pythagorean theorem to

\triangle ADC

using the known hypotenuse.

AD^2 + CD^2 = AC^2

, which becomes

AD^2 + 15^2 = 25^2

.

The hypotenuse

AC

is shared, and leg

CD

is given as 15, leaving

AD

as the only unknown variable.

4

Solve the equation for the missing leg

AD

.

AD^2 + 225 = 625 \Rightarrow AD^2 = 400 \Rightarrow AD = 20

.

Isolating

AD^2

and taking the square root provides the final length of the side.

Anahtar Kavram

Applying the Pythagorean theorem sequentially across adjacent right triangles that share a common hypotenuse.

Tahmini Süre:2m 0s

Soru 6Soru →

An isosceles triangle has a perimeter of $36$ centimeters. The base of the triangle measures $10$ centimeters. What is the area of the triangle in square centimeters?

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

60

To find the area, you must first know the perpendicular height. By splitting the isosceles triangle in half via its altitude, a right triangle is formed with a hypotenuse of

13

(the side length) and a base of

5

(half of the full base). The Pythagorean theorem (

5^2 + h^2 = 13^2

) reveals the height is

12

. The full area is then calculated as

\frac{1}{2} \times \text{base} \times \text{height}

, which is

\frac{1}{2} \times 10 \times 12 = 60

.

Adım Adım Çözüm

1

Calculate the length of the two equal sides.

The two equal sides each measure

13

centimeters.

The perimeter is

36

cm and the base is

10

cm. The remaining

26

cm must be divided equally between the two identical legs:

26 \div 2 = 13

cm.

2

Determine the base of the right triangle formed by the altitude.

The base of the right triangle is

5

centimeters.

The altitude of an isosceles triangle bisects the base, splitting the

10

-centimeter base into two equal

5

-centimeter segments.

3

Calculate the height using the Pythagorean theorem.

The height is

12

centimeters.

In the right triangle formed, the hypotenuse is

13

and one leg is

5

. Let

h

be the height.

5^2 + h^2 = 13^2 \Rightarrow 25 + h^2 = 169 \Rightarrow h^2 = 144 \Rightarrow h = 12

.

4

Calculate the total area of the triangle.

The area is

60

square centimeters.

The area formula is

\frac{1}{2} \times \text{base} \times \text{height}

. Thus,

\frac{1}{2} \times 10 \times 12 = 60

.

Anahtar Kavram

Using the Pythagorean theorem to find missing dimensions in isosceles triangles to calculate area.

Soru 7Soru →

A guy wire is attached to the top of a $12$ -foot vertical pole. The wire is anchored to the ground $5$ feet from the base of the pole. What is the length of the guy wire?

13

feet

\sqrt{119}

feet

17

feet

169

feet

Cevabı ve açıklamayı göster

Cevap: $13$ feet

Cevap

The guy wire is

13

feet long.

The length of the guy wire represents the hypotenuse of a right triangle formed by the pole and the ground. According to the Pythagorean theorem (

a^2 + b^2 = c^2

), the square of the hypotenuse equals the sum of the squares of the two legs. Substituting the given values:

12^2 + 5^2 = 144 + 25 = 169

. Taking the square root of

169

gives

13

feet.

Adım Adım Çözüm

1

Identify the known values representing the legs of the right triangle.

Leg

a = 5

feet, Leg

b = 12

feet.

The pole and the ground meet at a right angle, so they form the legs of the triangle.

2

Apply the Pythagorean theorem (

a^2 + b^2 = c^2

).

5^2 + 12^2 = c^2

The theorem connects the lengths of the legs to the hypotenuse (the guy wire).

3

Square the lengths of the legs and sum them.

25 + 144 = 169

This calculates the value of the hypotenuse squared (

c^2

).

4

Take the square root of the sum.

c = \sqrt{169} = 13

This isolates

c

and provides the actual length of the hypotenuse.

Anahtar Kavram

The Pythagorean theorem is used to find the hypotenuse of a right triangle when both legs are known.

Soru 8Soru →

A vertical flagpole that is originally $32$ feet tall breaks at a certain height due to strong winds. The top part of the pole folds over without completely detaching, and the tip touches the level ground exactly $16$ feet away from the base of the pole. At what height from the ground did the flagpole break?

8 feet

12 feet

16 feet

20 feet

Cevabı ve açıklamayı göster

Cevap: 12 feet

Cevap

12 feet

The scenario creates a right triangle where the vertical leg is the standing part of the pole, the horizontal leg is the ground, and the hypotenuse is the broken top part. Letting the break height be

x

, the hypotenuse becomes

(32 - x)

. Using the Pythagorean theorem:

x^2 + 16^2 = (32 - x)^2

. Expanding the right side gives

x^2 + 256 = 1024 - 64x + x^2

. Subtracting

x^2

from both sides simplifies the equation to

256 = 1024 - 64x

. Adding

64x

and subtracting

256

gives

64x = 768

. Dividing by

64

yields

x = 12

. Therefore, the pole broke

12

feet above the ground.

Adım Adım Çözüm

1

Define the variables for the sides of the right triangle formed by the broken flagpole.

Let the height of the break (standing part) be

x

. The length of the fallen part (hypotenuse) is the remainder of the pole, which is

(32 - x)

. The base distance along the ground is

16

feet.

Establishing these algebraic relationships is necessary to apply the Pythagorean theorem to the scenario.

2

Set up the Pythagorean theorem equation (

a^2 + b^2 = c^2

) using the defined variables.

x^2 + 16^2 = (32 - x)^2

The standing pole and the ground form a right angle, making the standing pole and ground distance the legs, and the fallen part the hypotenuse.

3

Expand both sides of the algebraic equation.

x^2 + 256 = 1024 - 64x + x^2

Expanding the squared binomial

(32 - x)^2

correctly is required to isolate the variable

x

.

4

Solve for

x

by combining like terms.

Subtract

x^2

from both sides to get

256 = 1024 - 64x

. Rearrange to

64x = 768

, and divide by

64

to find

x = 12

.

Isolating

x

provides the exact height from the ground where the flagpole snapped.

Anahtar Kavram

Applying the Pythagorean Theorem with algebraic expressions

Soru 9Soru →

A hiker walks $8$ miles due north, then turns and walks $15$ miles due east. What is the straight-line distance, in miles, from the hiker's starting point to their current location?

Cevabı ve açıklamayı göster

Cevap: 17

Cevap

17

Because the hiker's path consists of two perpendicular segments (north and east), they form the legs of a right triangle measuring

8

miles and

15

miles. The straight-line distance from the start to the end is the hypotenuse of this triangle. Applying the Pythagorean theorem (

a^2 + b^2 = c^2

), we get

8^2 + 15^2 = c^2

, which simplifies to

64 + 225 = 289

. The square root of

289

is

17

. Therefore, the straight-line distance is

17

miles.

Adım Adım Çözüm

1

Visualize the geometrical shape formed by the hiker's path.

A right triangle is formed with legs of

8

miles and

15

miles.

The directions 'due north' and 'due east' are perpendicular, creating a

90^\circ

angle.

2

Set up the Pythagorean theorem equation.

8^2 + 15^2 = c^2

The straight-line distance corresponds to the hypotenuse (

c

) of the right triangle.

3

Square the values of the legs and add them together.

64 + 225 = 289

To evaluate the left side of the equation.

4

Calculate the square root of the sum.

c = \sqrt{289} = 17

To solve for

c

and find the final straight-line distance.

Anahtar Kavram

Calculating distance using the Pythagorean theorem

Soru 10Soru →

Read the geometric scenario below and calculate the missing perimeter. Enter only the numerical value.

Aşağıdaki boşlukları doldurun

In

\triangle PQR

, an altitude

PS

is drawn to the base

QR

, creating two right triangles,

\triangle PQS

and

\triangle PSR

. If the length of side

PQ

is

13

units, the length of side

PR

is

15

units, and the altitude

PS

is

12

units, the perimeter of the entire

\triangle PQR

is units.

Cevabı ve açıklamayı göster

Cevap

42

To find the perimeter of

\triangle PQR

, we need the lengths of its three outer sides:

PQ

,

PR

, and

QR

. We are given

PQ = 13

and

PR = 15

. The altitude

PS

divides

QR

into two segments,

QS

and

SR

, forming two right triangles. Using the Pythagorean theorem in

\triangle PQS

, we find

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

. Using the Pythagorean theorem in

\triangle PSR

, we find

SR = \sqrt{15^2 - 12^2} = \sqrt{81} = 9

. The total base is

QR = QS + SR = 5 + 9 = 14

. The perimeter is the sum of the outer sides:

13 + 15 + 14 = 42

.

Adım Adım Çözüm

1

Use the Pythagorean theorem in right

\triangle PQS

to find the length of base segment

QS

.

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

The altitude

PS

forms a right angle with the base, making

PQ

the hypotenuse. The missing leg is found by subtracting the square of the altitude from the square of the hypotenuse.

2

Use the Pythagorean theorem in right

\triangle PSR

to find the length of base segment

SR

.

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

Similarly,

PR

is the hypotenuse of the second right triangle, so the missing leg is found by subtracting the square of the altitude from the square of this hypotenuse.

3

Calculate the total length of the base

QR

.

QR = QS + SR = 5 + 9 = 14

The entire base

QR

is the sum of the two segments created by the altitude.

4

Calculate the perimeter of the full

\triangle PQR

.

Perimeter =

PQ + PR + QR = 13 + 15 + 14 = 42

The perimeter is the total length of the outer boundary of the triangle, which is the sum of its three main sides.

Anahtar Kavram

Applying the Pythagorean theorem in multi-step geometric figures to find missing side lengths and perimeter.

Tahmini Süre:2m 0s

Soru 11Soru →

The hypotenuse of a right triangle measures $13$ inches, and one of its legs measures $5$ inches. What is the length of the other leg?

12

inches

144

inches

194

inches

8

inches

Cevabı ve açıklamayı göster

Cevap: $12$ inches

Cevap

The length of the other leg is

12

inches.

The correct answer is found by properly substituting the hypotenuse (

13

) and the known leg (

5

) into the Pythagorean theorem (

a^2 + b^2 = c^2

). Rearranging to solve for the missing leg gives

a^2 = 13^2 - 5^2

, which simplifies to

a^2 = 169 - 25 = 144

. Taking the square root of

144

yields

12

.

Adım Adım Çözüm

1

Set up the Pythagorean theorem equation (

a^2 + b^2 = c^2

) with the given values.

a^2 + 5^2 = 13^2

The Pythagorean theorem relates the two legs (

a

and

b

) to the hypotenuse (

c

) of a right triangle.

2

Square the known numerical values.

a^2 + 25 = 169

Evaluating the exponents is the first operational step in solving the equation.

3

Isolate the variable term

a^2

by subtracting

25

from both sides.

a^2 = 144

This leaves the squared variable isolated on one side of the equation.

4

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

a = 12

The square root is the inverse operation of squaring, yielding the final length.

Anahtar Kavram

The Pythagorean theorem (

a^2 + b^2 = c^2

) can be rearranged to find a missing leg by subtracting the square of the known leg from the square of the hypotenuse (

a^2 = c^2 - b^2

).

Soru 12Soru →

A computer monitor is shaped like a rectangle. It has a diagonal measurement of $26$ inches and a height of $10$ inches. What is the perimeter of the computer monitor?

68

inches

240

inches

52

inches

72

inches

Cevabı ve açıklamayı göster

Cevap: $68$ inches

Cevap

The correct perimeter of the monitor is

68

inches.

The correct answer is found by first recognizing that the diagonal, height, and width form a right triangle. Using the Pythagorean theorem, the missing width is calculated as the square root of (

26^2 - 10^2

), which equals

24

inches. The perimeter of the rectangular monitor is then

2

times the height plus

2

times the width:

2(10) + 2(24) = 68

inches.

Adım Adım Çözüm

1

Identify the right triangle formed by the monitor's dimensions.

The diagonal (

26

inches) is the hypotenuse, and the height (

10

inches) is one leg.

A rectangle's diagonal divides it into two identical right triangles.

2

Use the Pythagorean theorem (

a^2 + b^2 = c^2

) to find the missing width.

10^2 + b^2 = 26^2

→

100 + b^2 = 676

→

b^2 = 576

→

b = 24

inches.

The width is required to calculate the perimeter of the rectangle.

3

Calculate the perimeter of the rectangle.

P = 2l + 2w = 2(10) + 2(24) = 20 + 48 = 68

inches.

The perimeter is the total distance around the outside of the monitor.

Anahtar Kavram

Triangles and Pythagorean Theorem

Soru 13Soru →

A theater crew is building a wooden stage prop in the shape of a right triangle. The vertical height of the prop is $12$ feet and the horizontal base is $5$ feet. What is the length of the slanted top edge of the prop?

169

feet

17

feet

13

feet

119

feet

Cevabı ve açıklamayı göster

Cevap: $13$ feet

Cevap

The length of the slanted top edge is

13

feet.

The slanted edge represents the hypotenuse of a right triangle. According to the Pythagorean theorem, the square of the hypotenuse equals the sum of the squares of the two perpendicular legs (

a^2 + b^2 = c^2

). Substituting the given legs (

5

and

12

), we calculate

5^2 + 12^2 = 25 + 144 = 169

. Taking the square root of

169

yields

13

feet.

Adım Adım Çözüm

1

Identify the given values as the legs of a right triangle.

The legs are

a = 5

and

b = 12

. The unknown slanted edge is the hypotenuse,

c

.

The Pythagorean theorem relates the two perpendicular legs to the longest side.

2

Substitute the leg values into the Pythagorean theorem formula,

a^2 + b^2 = c^2

.

5^2 + 12^2 = c^2

This sets up the equation to solve for the missing hypotenuse.

3

Square the numbers and add them together.

25 + 144 = c^2

, which simplifies to

169 = c^2

.

This isolates

c^2

on one side of the equation.

4

Take the square root of both sides to find the value of

c

.

c = 13

Taking the square root cancels the exponent and gives the final length of the edge.

Anahtar Kavram

Using the Pythagorean theorem to calculate the hypotenuse of a right triangle.

Soru 14Soru →

A rectangular garden gate needs a diagonal wooden brace for support. The gate is $4$ feet tall and $3$ feet wide. What is the length, in feet, of the diagonal brace?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

The diagonal brace forms the hypotenuse of a right triangle where the width (

3

feet) and height (

4

feet) are the legs. According to the Pythagorean theorem (

a^2 + b^2 = c^2

), we substitute the given lengths:

3^2 + 4^2 = c^2

. This simplifies to

9 + 16 = 25

. Taking the square root of

25

gives a diagonal length of

5

feet. This forms a common

3-4-5

Pythagorean triple.

Adım Adım Çözüm

1

Identify the right triangle formed by the gate

The legs of the right triangle are

3

and

4

, and the hypotenuse is the unknown diagonal brace.

The corners of a rectangular gate form right angles, so drawing a diagonal creates a right triangle.

2

Apply the Pythagorean theorem

3^2 + 4^2 = c^2

The Pythagorean theorem (

a^2 + b^2 = c^2

) is used to find the missing side length of a right triangle.

3

Calculate the squares of the legs

9 + 16 = c^2

Squaring the given lengths prepares the equation for addition.

4

Solve for the hypotenuse

25 = c^2

, which means

c = 5

Taking the square root of the sum yields the length of the diagonal brace.

Anahtar Kavram

Using the Pythagorean theorem to find the hypotenuse of a right triangle.

Soru 15Soru →

In a right triangle, the two shorter sides (legs) measure $10$ inches and $24$ inches. What is the length of the hypotenuse in inches?

34

60

26

676

Cevabı ve açıklamayı göster

Cevap: 26

Cevap

The length of the hypotenuse is 26 inches.

The hypotenuse is found by applying the Pythagorean theorem (

a^2 + b^2 = c^2

). When you square the given legs (

10

and

24

), you get

100

and

576

. Adding these gives

676

. Finally, taking the square root of

676

results in

26

. This is also recognized as a scaled version of the fundamental 5-12-13 Pythagorean triple (multiplied by 2).

Adım Adım Çözüm

1

Recall the Pythagorean theorem formula.

a^2 + b^2 = c^2

This formula connects the lengths of the legs (

a

and

b

) to the hypotenuse (

c

) in a right triangle.

2

Substitute the given leg lengths into the formula.

10^2 + 24^2 = c^2

The lengths of the two shorter sides are given as 10 and 24.

3

Square the lengths of the legs.

100 + 576 = c^2

Following the order of operations, evaluate the exponents before adding.

4

Add the squared values together.

676 = c^2

Combine the values to find the square of the hypotenuse.

5

Take the square root of both sides to solve for c.

c = \sqrt{676} = 26

Taking the square root provides the final length of the hypotenuse.

Anahtar Kavram

Pythagorean Theorem

Soru 16Soru →

Calculate the specified length for the given shape.

Aşağıdaki boşlukları doldurun

A student cuts a piece of cardboard into a right triangle. The horizontal base measures

15

cm and the vertical height measures

20

cm. The length of the slanted side (hypotenuse) is cm.

Cevabı ve açıklamayı göster

Cevap

25

Applying the Pythagorean theorem (

a^2 + b^2 = c^2

), we substitute the legs

a = 15

and

b = 20

. Squaring these gives

225

and

400

. Adding them yields

625

. Taking the square root of

625

gives the hypotenuse length of

25

cm. Alternatively, this can be recognized as a scaled

3-4-5

right triangle, where each side is multiplied by

5

(

3 \times 5 = 15

,

4 \times 5 = 20

, and

5 \times 5 = 25

).

Adım Adım Çözüm

1

Identify the known values in the Pythagorean theorem formula,

a^2 + b^2 = c^2

.

The legs are

a = 15

and

b = 20

. The hypotenuse

c

is unknown.

The Pythagorean theorem defines the relationship between the lengths of the two legs of a right triangle and its hypotenuse.

2

Substitute the leg values into the theorem and square them.

15^2 + 20^2 = 225 + 400 = 625

. Thus,

c^2 = 625

.

This isolates the square of the unknown hypotenuse (

c^2

) on one side of the equation.

3

Take the square root of both sides to find the hypotenuse.

c = \sqrt{625} = 25

.

The square root gives the actual length of the side rather than its squared value.

Anahtar Kavram

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs (

a^2 + b^2 = c^2

).

Soru 17Soru →

A guy wire is anchored to the flat ground and attached to a vertical pole, forming a right triangle. The wire is $10$ meters long and attaches to the pole exactly $8$ meters above the ground. What is the horizontal distance from the base of the pole to the anchor point on the ground, in meters?

6

2

18

\sqrt{164}

Cevabı ve açıklamayı göster

Cevap: $6$

Cevap

The correct answer is

6

.

The scenario describes a right triangle where the hypotenuse is

10

and one leg is

8

. Applying the Pythagorean theorem (

a^2 + b^2 = c^2

), we find the unknown leg distance is

\sqrt{10^2 - 8^2} = \sqrt{100 - 64} = \sqrt{36} = 6

.

Adım Adım Çözüm

1

Identify the known sides of the right triangle.

The hypotenuse (the wire) is

10

meters, and one vertical leg (the height on the pole) is

8

meters.

Assigning the correct values to the Pythagorean theorem formula ensures the proper calculation path.

2

Set up the Pythagorean theorem equation.

x^2 + 8^2 = 10^2

, where

x

is the unknown horizontal distance.

The Pythagorean theorem relates the lengths of the legs to the hypotenuse in a right triangle.

3

Square the known values and solve for

x^2

.

x^2 + 64 = 100

, which simplifies to

x^2 = 100 - 64 = 36

.

Isolating the squared variable term is necessary to find the missing side length.

4

Take the square root to find the final distance.

x = \sqrt{36} = 6

.

Taking the square root converts the squared area value back into a linear distance.

Anahtar Kavram

The Pythagorean theorem (

a^2 + b^2 = c^2

) allows you to find a missing leg of a right triangle when the hypotenuse and the other leg are known.

Soru 18Soru →

A sail on a small boat is shaped like a right triangle. The horizontal bottom edge of the sail is $7$ feet long, and the vertical edge is $24$ feet high. What is the length, in feet, of the longest edge (the hypotenuse) of the sail?

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

25

The correct answer is found by applying the Pythagorean Theorem (

a^2 + b^2 = c^2

). Substituting the lengths of the two legs (

7

and

24

), the equation becomes

7^2 + 24^2 = c^2

. Squaring the numbers yields

49 + 576

. Adding these together gives

625 = c^2

. Finally, taking the square root of

625

results in

25

, which represents the length of the hypotenuse.

Adım Adım Çözüm

1

Identify the known sides of the right triangle.

Leg

a = 7

feet and leg

b = 24

feet.

The horizontal and vertical edges of the sail represent the two perpendicular legs of a right triangle.

2

Set up the Pythagorean Theorem equation.

7^2 + 24^2 = c^2

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

c

).

3

Square the lengths of the legs.

49 + 576 = c^2

To evaluate the equation, the individual lengths must be squared before adding them.

4

Add the squared values together.

625 = c^2

Combining the squared leg lengths yields the square of the hypotenuse.

5

Calculate the square root to solve for c.

c = 25

Taking the square root of

625

gives the final length of the hypotenuse.

Anahtar Kavram

Pythagorean Theorem

Soru 19Soru →

A drone takes off from a launch pad and flies $8$ kilometers due north. It then turns and flies $15$ kilometers due east. What is the straight-line distance from the launch pad to the drone's current position?

13

kilometers

17

kilometers

23

kilometers

60

kilometers

Cevabı ve açıklamayı göster

Cevap: $17$ kilometers

Cevap

The straight-line distance is

17

kilometers.

The drone's path forms a right triangle because north and east are perpendicular directions. The straight-line distance between the launch pad and the current position is the hypotenuse. Using the Pythagorean theorem, we calculate

8^2 + 15^2 = 289

. The square root of

289

is

17

, making the straight-line distance

17

kilometers.

Adım Adım Çözüm

1

Identify the shape formed by the drone's flight path.

The path forms a right triangle, with the north and east paths as the legs.

North and east directions are perpendicular to each other, creating a

90^\circ

angle.

2

Set up the Pythagorean theorem equation.

a^2 + b^2 = c^2

, where

a = 8

and

b = 15

.

The straight-line distance from the start to the current position represents the hypotenuse (

c

).

3

Square the lengths of the two legs and add them together.

8^2 + 15^2 = 64 + 225 = 289

.

This calculates the value of

c^2

according to the theorem.

4

Find the square root of the sum to determine the hypotenuse.

\sqrt{289} = 17

.

Taking the square root of

c^2

gives the actual straight-line distance.

Anahtar Kavram

The Pythagorean theorem (

a^2 + b^2 = c^2

) is used to find the hypotenuse of a right triangle when both legs are known.

Soru 20Soru →

Read the scenario below and provide the missing numeric value.

Aşağıdaki boşlukları doldurun

An architect is designing a courtyard in the shape of a right triangle. The two perpendicular walkways that form the right angle measure

9

meters and

12

meters in length. The length of the diagonal path connecting their endpoints is meters.

Cevabı ve açıklamayı göster

Cevap

The diagonal path is

15

meters long.

By applying the Pythagorean theorem (

a^2 + b^2 = c^2

) with legs of

9

and

12

, we calculate

9^2 + 12^2 = 81 + 144 = 225

. The square root of

225

is

15

, which is the exact length of the diagonal path.

Adım Adım Çözüm

1

Identify the known sides of the right triangle.

The two perpendicular walkways represent the legs of the triangle (

a

and

b

), measuring

9

and

12

.

To accurately apply the Pythagorean theorem, the given measurements must be identified as either legs or the hypotenuse.

2

Apply the Pythagorean theorem (

a^2 + b^2 = c^2

).

9^2 + 12^2 = c^2

The theorem is used to find the hypotenuse (

c

) when the lengths of both legs are known.

3

Square the lengths of both legs.

81 + 144 = c^2

Following mathematical order of operations by evaluating the exponents first.

4

Add the squared values and solve for

c

.

225 = c^2

, which means

c = \sqrt{225} = 15

.

Taking the square root of the sum yields the final straight-line length of the hypotenuse.

Anahtar Kavram

Using the Pythagorean theorem (

a^2 + b^2 = c^2

) to calculate the length of the hypotenuse when both legs are known.

Triangles and Pythagorean Theorem

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