Geometry

1

The Pythagorean

Theorem

The Pythagorean Theorem

2

A

B

C

Given any right triangle, A2 + B2 = C2

Example3

A

B

C

In the following figure if A = 3 and B = 4,

Find C.

A2 + B2 = C2

32 + 42 = C 2

9 + 16 = C2

5 = C

Verifying the Pythagorean Theorem

4

Given a piece of graph paper, make a right triangle. Then make squares of the right triangle. Then find the square’s areas.

Pythagorean Theorem : Examples for finding the hypotenuse.

5

A=8, B= 15, Find C

A=7, B= 24, Find C

A=9, B= 40, Find C

A=10, B=24, Find C

A =6, B=8, Find C

A

B

C

C = 17

C = 25

C = 41

C = 26

C = 10

Finding the legs of a right triangle:

6

A

B

C

In the following figure if B = 5 and C = 13,

Find A.

A2 + B2 = C2

A2 +52 = 132

A2 + 25 = 169

A2 = 144

A = 12

More Examples:7

1) A=8, C =10 , Find B2) A=15, C=17 , Find B3) B =10, C=26 , Find A4) A=15, B=20, Find C5) A =12, C=16, Find B6) B =5, C=10, Find A7) A =6, B =8, Find C8) A=11, C=21, Find B

A

B

C

B = 6

B = 8A = 24

C = 25B = 10.6A = 8.7C = 10B = 17.9

Given the lengths of three sides, how do you know if you have a right triangle?

8

A

B

C

Given A = 6, B=8, and C=10, describe the triangle.

A2 + B2 = C2

62 +82 = 102

36 + 64 = 100

* This is true, so you have a right triangle.

Pythagorean Triples

9

Some right-angled triangles where all three sides are whole numbers called Pythagorean Triangles.

The three whole number side-lengths are called a Pythagorean triple.

•The 3-4-5 triangle • An example is a = 3, b = 4 and h = 5, called "the 3-4-5 triangle". We can check it as follows:

Pythagorean TriplesNot only is 3-4-5 a Pythagorean triple, but

so is any multiple of 3-4-5. 3-4-5 6-8-10 12-16-20 15-20-25 18-24-30

10

Can you think of any others??

5,12,13 7,24,2520,21,29

What happens if it the Pythagorean Theorem does NOT work? If you do not have a picture nor an

angle that you know for a fact is 90 degrees, then it is possible to have an acute or an obtuse triangle.

If A2 + B2 > C2, you have an acute triangle.

If A2 + B2 < C2, you have an obtuse triangle.

11

If A2 + B2 > C2, it is an acute triangle.

12

Given A = 4, B = 5, and C = 6, describe the triangle.

A2 + B2 = C2

42 + 52 = 62

16 + 25 = 36

41 > 36, so we have an acute triangle.

A B

C

If A2 + B2 < C2, it is an obtuse triangle.

13

Given A = 4, B = 6, and C =8, describe the triangle.

A2 + B2 = C2

42 + 62 = 82

16 + 36 = 64

52 < 64, so we have an obtuse triangle.

A

C

B

Describe the following triangles asacute, right, or obtuse

14

1) A=9, B=40, C=412) A=10, B=15, C=203) A=2, B=5, C=64) A=12, B=16, C=205) A=11, B=12, C=146) A=2, B=3, C=47) A=1, B=7, C=78) A=90, B=120,

C=150

A

B

C

right

acute

obtuse

right

right

right

obtuse

acute