The Pythagorean Theorem

Parts of a Right Triangle Hypotenuse Leg Leg

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.

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

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 __

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

Example __ __ x 12

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)

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

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.

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 52.95 meters.

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?

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

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

If c2<a2+b2, then the Δ is acute.

If c2>a2+b2, then the Δ is obtuse.

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) 3.2+4.8>5.1; 4.8+5.1>3.2; and 3.2+5.1>4.8 What kind of Δ? 5.12___4.82+3.22 26.01___23.04+10.24 26.01___33.28 acute <

Assignment