﻿ Converse of the Pythagorean Theorem Worksheet - Page 2 | Problems & Solutions

# Converse of the Pythagorean Theorem Worksheet - Page 2

Converse of the Pythagorean Theorem Worksheet
• Page 2
11.
Find the value of $x$ in the right triangle.

 a. 12 b. 16 c. 13 d. 3

#### Solution:

a2 + b2 = c2
[Write Pythagorean theorem.]

x2 + 92 = (x + 3)2
[Substitute for a, b and c.]

x2 + 81 = x2 + 9 + 6x
[Simplify.]

72 = 6x
[Subtract (x2 + 9) from both sides.]

x = 726
[Divide each side by 6.]

= 12

12.
Find the value of $x$ in the figure.

 a. 30.25 b. 36.24 c. 28.125 d. 35.8

#### Solution:

The triangle is a right triangle.

152 + (x - 2)2 = (x + 2)2
[Use Pythagorean theorem.]

225 + x2 + 4 - 4x = x2 + 4 + 4x
[Apply exponents.]

225 = 8x
[Add 4x to both sides and simplify.]

x = 2258
[Divide each side by 8.]

= 28.125

The value of x is 28.125.

13.
A ladder which is 15 feet long is placed on a wall such that the top of the ladder touches the top of the wall. The bottom of the ladder is 9 feet away from the wall. What is the height of the wall?
 a. 15 feet b. 18 feet c. 12 feet d. 9 feet

#### Solution:

The length of the ladder l = 15 feet.

The distance from the foot of the ladder to the wall, d = 9 feet.

Let h be the height of the wall.

d2 + h2 = l2
[Write Pythagorean theorem.]

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

= 152 - 92
[Substitute l and h.]

= 225 - 81
[Apply exponents and simplify.]

= 144

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

= 12

Height of the wall = 12 feet.

14.
State whether the lengths 9, 10 and 11 are sides of a right triangle.
 a. data insufficient b. yes c. cannot say d. no

#### Solution:

112 ≠ 92 + 102
[Check for Pythagorean theorem.]

The lengths 9, 10 and 11 do not satisfy the Pythagorean theorem.

The lengths 9, 10 and 11 are not sides of a right triangle.

15.
Find the measures of the sides of the right triangle shown.

 a. 3, 4 and 6 b. 5, 12 and 13 c. 4, 11 and 13 d. 6, 8 and 10

#### Solution:

a2 + b2 = c2
[Write Pythagorean theorem.]

(2x)2 + (3x - 1)2 = (3x + 1)2
[Substitute for a, b and c.]

4x2 + 9x2 + 1 - 6x = 9x2 + 1 + 6x

4x2 = 12x
[Subtract (9x2 + 1 - 6x) from both the sides.]

x = 12x4x = 3
[Divide each side by 4x.]

2x = 2(3) = 6
[Substitute x = 3.]

3x - 1 = 3(3) - 1 = 8
[Substitute x = 3.]

3x + 1 = 3(3) + 1 = 10
[Substitute x = 3.]

The measures of the sides of the given right triangle are 6, 8 and 10.

16.
Find the length of AC in the dot paper, if the distance between each dot is one unit.

 a. 10$\sqrt{5}$ units b. 8$\sqrt{5}$ units c. 5$\sqrt{5}$ units d. 8 units

#### Solution:

From the figure, AB = 11 units and BC = 2 units.

AC2 = AB2 + BC2
[Apply Pythagorean theorem.]

AC2 = 112 + 22
[Substitute the values of AB and BC.]

AC2 = 121 + 4 = 125
[Apply exponents and simplify.]

AC = 125 = 25 × 5 = 55
[Take square root of both sides.]

The length of AC is 55 units.

17.
What are the side measures of the blue colored right triangle enclosed in the dot paper, if the distance between each dot is one unit?

 a. 3, 2 and 5 b. 5, 6 and 9 c. 10, 11 and 18 d. 3, 2 and $\sqrt{13}$

#### Solution:

The lengths of two legs of right triangle in the figure are 3 units and 2 units.

(Hypotenuse)2 = 32 + 22
[Apply Pythagorean theorem.]

= 9 + 4
[Apply exponents.]

= 13

Hypotenuse = 13
[Take square root of both sides.]

The side measures of the right triangle are 3 units, 2 units and 13 units.

18.
What are the measures of the sides of right triangle AED in the figure, if the distance between two dots in the figure is one unit?

 a. 3, 8 and 25 b. 2, 3 and 5 c. 2, 3 and $\sqrt{13}$ d. 4, 9 and 13

#### Solution:

From the figure, AE = 2 units and DE = 3 units and AD is the hypotenuse.

[Apply Pythagorean theorem.]

= 22 + 32
[Substitute the values of AE and DE.]

= 4 + 9
[Apply exponents and simplify.]

= 13

[Take square root on both sides.]

The side measures of the right triangle are 2 units, 3 units and 13 units.

19.
What is the length of the third side of the triangle shown in the figure?

 a. 4 b. $\sqrt{15}$ c. $\sqrt{34}$ d. $\sqrt{8}$

#### Solution:

One angle of the triangle is 90o. So, the triangle is a right triangle.

The side opposite to the right angle is hypotenuse.

Let p be the hypotenuse.

Applying Pythagorean theorem, p2 = 32 + 52

p2 = 9 + 25 = 34

p = 34
[Take square root on both the sides.]

The third side of the triangle is 34.

20.
What are the values of $x$ and $y$ in the triangle?

 a. 4 units and 15 units b. 2$\sqrt{5}$ units and 45 units c. $\sqrt{160}$units and $\sqrt{41}$ units d. 24 units and 45 units

#### Solution:

ΔBDC is a right triangle.

BC2 = BD2 + CD2
[Apply Pythagorean theorem.]

x2 = 122 + 42
[Substitute BC, BD and CD.]

x2 = 144 + 16 = 160
[Evaluate powers and simplify.]

x = 160
[Take square root both sides.]

[Apply Pythagorean theorem.]

y2 = 52 + 42