﻿ Converse of the Pythagorean Theorem Worksheet | Problems & Solutions

# Converse of the Pythagorean Theorem Worksheet

Converse of the Pythagorean Theorem Worksheet
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1.
Figure ABCD is a rectangular field 400 ft long and 300 ft wide. The path from A to C is how much shorter than the path from A to B to C?

 a. 400 ft b. 700 ft c. 200 ft d. 300 ft

#### Solution:

Length of the rectangular field ABCD is AB = CD = 400 ft.

Width of the rectangular field ABCD is BC = AD = 300 ft.

Let p be the distance between A to C.

According to Pythagorean theorem p2 = 4002 + 3002.

p2 = 160000 + 90000

p2 = 250000

p2 = 250000

p = 500

Path from A to C = distance between A to C = 500 ft.

Total path from A to B to C = 400 + 300 = 700 ft.

Total path from A to B to C - Path from A to C = 700 - 500 = 200 ft.

The path from A to C is 200 ft shorter than the total path from A to B to C.

2.
Find the length of the side PR of ΔPQR shown in the graph.

 a. $\sqrt{20}$ units b. $\sqrt{35}$ units c. $\sqrt{78}$units d. $\sqrt{89}$ units

#### Solution:

In the graph, ΔPQR is a right triangle.

From the graph, PQ = 8 units and QR = 5 units

According to Pythagorean theorem, in a right triangle, square of the hypotenuse = sum of the squares of the other two sides.

In ΔPQR, PR is hypotenuse.

From ΔPQR, PR2 = PQ2 + QR2

PR2 = 82 + 52 = 64 + 25 = 89

PR = 89

The length of the side PR = 89 units.

3.
The numbers 3, 4 and 5 are a Pythagorean triplet because 32 + 42 = 52. What are the next two multiples of this Pythagorean triplet?
 a. (5, 12 and 13), (9, 12 and 15) b. (6, 8 and 10), (5, 12 and 13) c. (6, 8 and 10), (9, 12 and 15) d. (6, 8 and 10), (9, 16 and 15)

#### Solution:

Next two multiples of Pythagorean triplet, 3, 4 and 5 = (6, 8 and 10), (9, 12 and 15)
[Multiply each number by 2 and 3.]

4.
The hypotenuse of a right triangle is 3 feet more than its longer leg. The length of shorter leg is 9 feet. Find the length of the longer leg and hypotenuse.
 a. 14 feet and 17 feet b. 13 feet and 15 feet c. 12 feet and 17 feet d. 12 feet and 15 feet

#### Solution:

Let s be the longer leg of the right triangle.

Hypotenuse is 3 feet more than the length of longer leg.

Hypotenuse = s + 3

The length of shorter leg = 9 feet.

s2 + 92 = (s + 3)2
[Apply Pythagorean theorem.]

s2 + 81 = s2 + 6s + 9
[Apply exponents.]

72 = 6s
[Subtract (s2 + 9) from both sides.]

s = 726
[Divide by 6 on both sides.]

= 12

Hypotenuse = s + 3

= 12 + 3
[Substitute s.]

= 15

Length of longer leg is 12 feet and length of hypotenuse is 15 feet.

5.
A right triangle never has two equal angles. State whether the statement is true or false.
 a. True b. False

#### Solution:

A right triangle can have two equal angles.

For example, a triangle with angle measures 45o, 45o and 90o is a right triangle.

So, the statement is false.

6.
One end of a wire with a length of 15 feet is tied to the top of the pole and the other end is fixed to the ground at a distance of 9 feet from the foot of the pole. What is the height of the pole?

 a. 21 feet b. 6 feet c. 12 feet d. 27 feet

#### Solution:

Let h be the height of the pole.

d2 + h2 = l2
[Apply Pythagorean theorem.]

h2 = l2 - d2
[Subtract d2 from both sides.]

h2 = 152 - 92
[Substitute l and d.]

= 225 - 81
[Apply exponents and simplify.]

= 144

h = 144
[Take square root of both sides.]

= 12

The height of the pole is 12 feet.

7.
One leg A of a right triangle is 1 inch less than the hypotenuse. The length of the other leg is 3 inches. What are the lengths of leg A and the hypotenuse?
 a. 3 inches and 4 inches b. 4 inches and 5 inches c. 6 inches and 7 inches d. 6 inches and 5 inches

#### Solution:

Let s be the length of the hypotenuse.

One leg A of right triangle is 1 inch less than the hypotenuse.

So, the length of leg A = s - 1

The length of another leg = 3 inches.

(s - 1)2 + 32 = s2
[Apply Pythagorean theorem]

s2 + 1 - 2s + 9 = s2

10 - 2s = 0
[Subtract s2 from both sides and simplify.]

10 = 2s

s = 102 = 5
[Divide by 2 on both sides.]

Length of the hypotenuse s = 5 inches.

= 5 - 1 = 4
Length of the leg A = s - 1
[Substitute s.]

Length of the hypotenuse is 5 inches and length of leg A is 4 inches.

8.
What is the measure of the hypotenuse in the right triangle?

 a. 25 b. 16 c. 10 d. 15

#### Solution:

From the figure, a = 15 and b = 20

c2 = a2 + b2
[Write Pythagorean theorem.]

c2 = 152 + 202
[Substitute for a and b.]

c2 = 225 + 400 = 625
[Apply exponents and simplify.]

c = 625 = 25
[Take square root of both sides and simplify.]

The measure of the hypotenuse, c = 25.

9.
What is the value of $b$ in the figure?

 a. 32 b. 36 c. 30 d. 35

#### Solution:

From the figure, a = 12 and c = 37

c2 = a2 + b2
[Write Pythagorean theorem.]

c2 - a2 = b2
[Subtract a2 from both sides.]

b2 = 372 - 122
[Substitute a and c values.]

b2 = 1369 - 144 = 1225
[Apply exponents and simplify.]

b = 1225 = 35
[Take square root of both sides.]

The value of b is 35.

10.
Find the value of $a$ from the figure.

 a. 12 b. 8 c. 16 d. 13

#### Solution:

From the figure, c = 17 and b = 15

a2 + b2 = c2
[Write Pythagorean theorem.]

a2 = c2 - b2
[Subtract b2 from both sides.]

a2 = 172 - 152
[Substitute for b and c.]

a2 = 289 - 225 = 64
[Apply exponents and simplify.]

a = 64 = 8
[Take square root of both sides.]

The value of a is 8.