CONFIDENTIAL 1

Grade 8 Pre-AlgebraGrade 8 Pre-Algebra

Pythagorean Pythagorean Theorem 2Theorem 2

Pythagorean Theorem

The Pythagorean Theorem shows how the legs and hypotenuse of a right triangle are related.

legs

hypotenuse

In a right triangle , the two shortest sides are legs.

The longest side, which is opposite the right angle, is the hypotenuse.

In words: In a right angled triangle, the square of the length of the hypotenuse is equal to the sum of the square of the lengths of the legs.

In Symbols: a2 +b2 = c2 .

a

b

c

If we know the lengths of two sides of a right angled triangle, then Pythagoras' Theorem allows

us to find the length of the third side.

Pythagorean Theorem

4

hypotenuse = 2 . shorter leglonger leg = shorter leg . √3

C

30º

60º

2s

A

B s

√3s

Using the Pythagorean theorem, we find that

The converse of Pythagorean Theorem allows you to substitute the lengths of the sides of a triangle into the

equation : c2 = a2 +b2

to check whether a triangle is a right triangle, if the Pythagorean equation is true the triangle is a right triangle.

Let us do some practice problems!

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1) Is every isosceles triangle an isosceles right triangle?

a) True

b) False

b) False

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2) A ladder that is 10 feet long leans against a building. The bottom of the ladder is 4 feet away from the base. How far up the side of the building

does the ladder reach?

a) 7.2 ftb) 10.2 ftc) 8.2 ftd) 9.2 ft

d) 9.2 ft

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3) If one side of a right triangle is 3 ft and the length of its hypotenuse is 4 ft, then find the length of the

other side. 

a) √3 ftb) √5 ftc) √7 ftd) √2 ft

c) √7 ft

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4) A ladder which is 25 ft 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 15 ft away

from the wall. What is the height of the wall?

a) 20 ftb) 15 ftc) 10 ftd) 16 ft

a) 20 ft

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5) If a = 12 cm and b = 14 cm, then find the value of c in the right triangle.

a) 15.63 cm b) 14.63 cm c) 18.43 cm d) 12.43 cm

c) 18.43 cm

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6) The vertices of 3 squares are joined to form a right triangle. What is the area of the

smallest square?

54 cm2

36 cm2

1) 18 cm2

2) 20 cm2

3) 22 cm2

4) 14 cm2

a) 18 cm2

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7) The floor mat used in gymnastics measures 40 feet by 40 feet. Gymnasts use the diagonal of the

mat for most of their tumbling patterns. How many feet are available to the gymnasts by using the diagonal of the mat for their tumbling patterns.

a) 49.2 ftb) 50.2 ftc) 66.6 ftd) 56.6 ft

d) 56.6 ft

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8) The length and the width of a rectangular football court are 40 feet and 45 feet respectively. A diagonal walkway is made from one end to the opposite end.

What is the length of the walkway?  

a) 61.2 ftb) 60.2 ftc) 59.2 ftd) None of the above

b) 60.2 ft

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9) The lengths of the perpendicular sides of a right angled triangle 3 cm. and 4 cm.

Then find the perimeter of the triangle.

d) 12 cm

a) 18 cmb) 16 cmc) 14 cmd) 12 cm

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10) One leg of a right triangle is 15 inches long. The other leg is 3 inches less than the hypotenuse. What are the lengths of this leg and hypotenuse?

a) 27 inb) 34 inc) 36 ind) 35 in

c) 36 in

Assessment

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1) Figure ABCD is a rectangular field 136 ft long and 102 ft wide. By how much is the path from A to

C shorter than the path from A to B to C?

a) 68 ftb) 66 ftc) 72 ftd) 70 ft

a) 68 ft

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2) The length and the width of a rectangular field are 28 feet and 21 feet respectively. A diagonal

walkway is made from one end to the opposite end. What is the length of the walkway?

a) 39 ftb) 37 ftc) 35 ftd) None of the above

c) 35 ft

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3) State which of the following measures will form a right triangle.

a) 12, 14, 20b) 24, 18, 30c) 12, 13, 15d) None of the above

b) 24, 18, 30

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4) The perimeter of a right triangle is 150 inches and the ratio of the lengths of the 2 legs is 5 : 12.

Find the length of the hypotenuse.

a) 70 inb) 75 inc) 60 ind) 65 in

d) 65 in

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5) Sheath went to a level field to fly a kite. She let out all 85 ft of string and tied it to a stake. Then she

walked out on the field until she was directly under the kite, 51 ft from the stake. How high was the kite?

a) 68 ftb) 70 ftc) 72 ftd) 66 ft

a) 68 ft

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6) What is the length of the third side of the triangle shown in the figure? [Given a = 6.6 in. and

b = 8.8 in. .]

6.6 in

8.8 in x

a) 12 inb) 10 inc) 11 ind) 13 in

c) 11 in

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7) What is the measure of the side of the square, if the diagonal of the square is 8 feet?

a) 3.7 ftb) 5.7 ftc) 4.7 ftd) 6.7 ft

b) 5.7 ft

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8) The lengths of the perpendicular sides of a right angled triangle 30 cm. and 16 cm. Then find the

perimeter of the triangle.

a) 75 cmb) 72 cmc) 78 cmd) 80 cm

d) 80 cm

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9) If one side of a right triangle is two times the other and the length of hypotenuse is 25 ft, then

what are the measures of the two sides?

a) 5√5 ftb) 4√5 ftc) 7√5 ftd) 6√5 ft

a) 5√5 ft

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10) The lengths of the hypotenuse and the shorter leg of a right triangle are in the ratio 5:3 and the perimeter of the triangle is 24 inches. Find the

length of the longer leg.

a) 7 inb) 8 inc) 8 ind) 8 in

b) 8 in