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Pythagorean theorem word problems

Pythagorean theorem word problems arise in numerous situations. We will cover a couple a solid examples here.

Pythagorean problem # 1

John leaves school to go home. He walks 6 blocks North and then 8 blocks west. How far is John from the school?

Here is how you can model this situation


The distance from school to home is the length of the hypotenuse.

Let c be the missing distance from school to home and a = 6 and b = 8

c2 = a2 + b2

c2 = 62 + 82

c2 = 36 + 64

c2 = 100

c = √100

c = 10

The distance from school to home is 10 blocks.

Pythagorean problem # 2

A 13 feet ladder is placed 5 feet away from a wall. The distance from the ground straight up to the top of the wall is 13 feet Will the ladder the top of the wall?


Let the length of the ladder represents the length of the hypotenuse or c = 13 and a = 5 the distance from the ladder to the wall.

c2 = a2 + b2

132 = 52 + b2

169 = 25 + b2

169 - 25 = 25 - 25 + b2 (minus 25 from both sides to isolate b2 )

144 = 0 + b2

144 = b2

b = √144 = 12

The ladder will never reach the top since it will only reach 12 feet high from the ground yet the top is 14 feet high.

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