Pythagorean Theorem.

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Presentation transcript:

Pythagorean Theorem

Review

c² = a² + b² 25 = 16 + 9

Pythagorean Theorem A right triangle is a triangle with a 90° angle.

Pythagorean Theorem The angles in any triangle add up to 180°. In any triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle. Thus, in a right triangle, the hypotenuse is always the longest side. Explain this reasoning (Think Pair Share)

Pythagorean Theorem In a right triangle, the side opposite the 90° angle is called the hypotenuse and the remaining two sides are called the legs. Hypotenuse Leg Leg

Pythagorean Theorem Question: Suppose that two angles in a triangle are 60° and 30°. Is it a right triangle? Solution: Yes. The third angle must be 180° - 60° - 30° = 90°.

Pythagorean Theorem Question: Suppose that a triangle has a 100° angle. Is it a right triangle? Answer YES, NO, or MAYBE. Solution: No. The remaining two angles must sum to 80°, so neither remaining angle is a 90° angle.

Pythagorean Theorem Question: Suppose that a triangle has a 70° angle. Is it a right triangle? Answer YES, NO, or MAYBE. Solution: Maybe. The remaining two angles must sum to 110°, so one of the remaining angles could be a 90° angle.

Pythagorean Theorem Question: Suppose that two angles in a triangle are 80° and 40°. Is it a right triangle? Solution: No. The third angle must be 180° - 80° - 40° = 60°.

Pythagorean Theorem the sum of the squares of the shorter sides is equal to the square of the hypotenuse. if a triangle has this kind of relationship between the lengths of its sides, then it must be a right triangle!

Pythagorean Theorem c² = a² + b²

Pythagorean Theorem Question: Suppose the legs of a right triangle have lengths 8 and 6 , Find the hypotenuse Solution: 82 + 62 = x2 64 + 36 = x2 100 = x2 √100 = √x2 x = 10

Pythagorean Theorem Question: Suppose the legs of a right triangle have lengths 3 and 6 , Find the hypotenuse Solution: 32 + 62 = x2 9 + 36 = x2 45 = x2 x = ~ 6.7

Pythagorean Theorem Question: Suppose the legs of a right triangle have lengths 3 and x , and the hypotenuse has length 5 . Find x . Solution: 32 + x2 = 52 9 + x2 = 25 x2 = 16 √x2 = √16 x = 4

Pythagorean Theorem Question: Suppose the legs of a right triangle have lengths 4 and x , and the hypotenuse has length 7 . Find x . Solution: 42 + x2 = 72 16 + x2 = 49 x2 = 33 √x2 = √33 x = ~5.74

Pythagorean Theorem Summary c2 = a2 + b2 (if looking for hypotenuse) c2 - a2 = b2 (if have hypotenuse) Workout Complete Worksheet Quiz next class