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

Correct answer : (1)
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.

Correct answer : (3)
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.

Correct answer : (3)
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.

Correct answer : (4)
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.

Correct answer : (4)
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.

Correct answer : (3)
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.

Correct answer : (4)
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.

AD2 = AE2 + DE2
[Apply Pythagorean theorem.]

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

= 4 + 9
[Apply exponents and simplify.]

= 13

AD = 13
[Take square root on both sides.]

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

Correct answer : (3)
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
[Apply exponents and add.]

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

The third side of the triangle is 34.

Correct answer : (3)
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.]

ΔADC is a right triangle.

AC2 = AD2 + CD2
[Apply Pythagorean theorem.]

y2 = 52 + 42
[Substitute AC, AD and CD.]

y2 = 25 + 16 = 41
[Evaluate powers and simplify.]

y = 41
[Take square root both sides.]

x = 160 units and y = 41 units.

Correct answer : (3)

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