Converse of the Pythagorean Theorem Worksheet

**Page 1**

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 |

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

Let

According to Pythagorean theorem

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.

Correct answer : (3)

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 |

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, PR

PR

PR =

The length of the side PR =

Correct answer : (4)

3.

The numbers 3, 4 and 5 are a Pythagorean triplet because 3^{2} + 4^{2} = 5^{2}. 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) |

[Multiply each number by 2 and 3.]

Correct answer : (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 |

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

Hypotenuse =

The length of shorter leg = 9 feet.

[Apply Pythagorean theorem.]

[Apply exponents.]

72 = 6

[Subtract (

[Divide by 6 on both sides.]

= 12

Hypotenuse =

= 12 + 3

[Substitute

= 15

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

Correct answer : (4)

5.

A right triangle never has two equal angles. State whether the statement is true or false.

a. | True | ||

b. | False |

For example, a triangle with angle measures 45

So, the statement is false.

Correct answer : (2)

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 |

[Apply Pythagorean theorem.]

[Subtract

[Substitute

= 225 - 81

[Apply exponents and simplify.]

= 144

[Take square root of both sides.]

= 12

The height of the pole is 12 feet.

Correct answer : (3)

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 |

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

So, the length of leg A =

The length of another leg = 3 inches.

(

[Apply Pythagorean theorem]

10 - 2

[Subtract

10 = 2

[Add 2

[Divide by 2 on both sides.]

Length of the hypotenuse

= 5 - 1 = 4

Length of the leg A =

[Substitute

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

Correct answer : (2)

8.

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

a. | 25 | ||

b. | 16 | ||

c. | 10 | ||

d. | 15 |

[Write Pythagorean theorem.]

[Substitute for

[Apply exponents and simplify.]

[Take square root of both sides and simplify.]

The measure of the hypotenuse,

Correct answer : (1)

9.

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

a. | 32 | ||

b. | 36 | ||

c. | 30 | ||

d. | 35 |

[Write Pythagorean theorem.]

[Subtract

[Substitute

[Apply exponents and simplify.]

[Take square root of both sides.]

The value of

Correct answer : (4)

10.

Find the value of $a$ from the figure.

a. | 12 | ||

b. | 8 | ||

c. | 16 | ||

d. | 13 |

[Write Pythagorean theorem.]

[Subtract

[Substitute for

[Apply exponents and simplify.]

[Take square root of both sides.]

The value of

Correct answer : (2)