1. The Pythagorean Theorem

2. Parts of a Right Triangle
Hypotenuse
Leg
Leg

3. PROVING THE PYTHAGOREAN THEOREM
The Pythagorean Theorem is one of the most famous theorems in mathematics. The relationship it describes has been known for thousands of years.

4. PROVING THE PYTHAGOREAN THEOREM
THEOREM 9.4 Pythagorean Theorem
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. b a c
c 2 = a 2 + b 2

5. The Pythagorean Theorem
In a rt Δ the square of the length of the hypot. is = to the sum of the squares of the lengths of the legs.
c2 = a2+b2 c a __ b __

6. Example
x2=72+242
x2=49+576
x2=625
x=25 x 7 __ __ 24

7. Example __ __ x 12

8. not a pythag. triple (32+72102)
Pythagorean triple
Pythagorean triple- a set of 3 positive integers that satisfy the pythag thm.
3, 4, 5
(32+42=52)
7, 24, 25
(72+242=252)
3, 7, 10
not a pythag. triple (32+72102)

9. Ex: Find the area of the Δ to the nearest tenth of a meter.
A= ½ bh
25+h2=64
h2=39 8m 8m h
A= ½ bh __ __
10m

10. SUPPORT BEAM These skyscrapers are connected
by a skywalk with support beams. You can use the Pythagorean Theorem to find the approximate length of each support beam.

11. 23.26 m
47.57 m x
support
beams
Each support beam forms the hypotenuse of a right triangle. The right triangles are congruent, so the support beams are the same length.
x 2 = (23.26)2 + (47.57)2
Pythagorean Theorem
x = (23.26)2 + (47.57)2
Find the square root.
x  52.95
Use a calculator to approximate.
The length of each support beam is about meters.

12. Ladder against a wall
An 8 ft ladder is leaning against a wall with its base 3 ft from the wall. How high is the ladder up the wall?

13. Converse of the Pythagorean Theorem
If c2=a2+b2, then Δ ABC is a right Δ. B c a A C b

14. Example Are the Δs right Δs?7 10 6
13.4 12
13.42=102+72
179.56=100+49
179.56=149
36*5=36+144
180=180

15. If c2<a2+b2, then the Δ is acute.

16. If c2>a2+b2, then the Δ is obtuse.

17. Example Can the given side lengths form a Δ and if so, what kind of Δ would it be?
3.2, 4.8, 5.1
Yes, they form a Δ (by the Δ inequal. Thm)
>5.1; >3.2; and >4.8
What kind of Δ?
5.12___
26.01___
26.01___33.28
acute
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18. Assignment