The Pythagorean Theorem

The Pythagorean Theorem

a² + b² = c²

Right Triangle

- a triangle with a right angle.

- longest side: hypotenuse (c)

- two shorter sides: legs (a or b)

Pythagoras

Pythagorean Theorem

If given the right triangle with legs a and b and hypotenuse c,

then a² + b² = c².

Example 1

a = 8cm

b = 6cm

c = ?

Solution:

a² + b² = c²

8² + 6² = c²

64 + 36= c²

100 = c²

c = 10 cm

Example 2

a = ?

b = 5

c = 13

Solution:

a² + b² = c²

a² + 5² = 13²

a² + 25= 169

a² = 144

a = 12

Example 3

a = 24
b = ?
c = 31

Solution:
 $a^2+b^2=c^2$  
 $24^2+b^2=31^2$  
 $576+b^2=961$  
 $b^2=385$  
 $b=\sqrt{385}$  

Multiple Choice

• $\sqrt{64}$

• 8

• 16

• 12

Fill in the Blank

Find the missing side: a = 8cm, b = 15cm, c = ?

Multiple Choice

• $\sqrt{29725}$

• $5\sqrt{1189}$

• $\sqrt{15275}$

• $5\sqrt{611}$

Fill in the Blank

A lighthouse, 25 meters high, has spotted a boat sailing 180 meters away from it. How far is the boat from the top of the lighthouse?

We need to find c:

 $c^2=a^2+b^2$  
 $c^2=25^2+180^2$  
 $c^2=625+32400$  
 $c^2=33025$  
 $c=5\sqrt{1321}\approx181.73\ m$  

Therefore, the boat is 181.73 meters away from the top of the lighthouse.

Fill in the Blank

The foot of a ladder is placed 6 feet from a wall. If the top of the ladder rests 8 feet up on the wall, how long is the ladder?