Bellringer Simplify each expression 5 ∙ 10 243 6 ∙ 8

7-2 The Pythagorean Theorem and Its Converse

Pythagorean Theorem (Thm Pythagorean Theorem (Thm. 7-4): In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

Pythagorean Theorem 𝑎 2 + 𝑏 2 = 𝑐 2 Leg Leg Hypotenuse

Pythagorean Triple: set of nonzero whole numbers, a, b, and c that satisfy the Pythagorean Theorem Common Triples: 3, 4, 5 5, 12, 13 7, 24, 25 8, 15, 17 12, 25, 37 20, 21, 29

𝑎 2 + 𝑏 2 = 𝑐 2 8 2 + 𝑏 2 = 12 2 64+ 𝑏 2 =144 𝑏 2 =80 𝑏 2 = 80 𝑏=4 5

Bellringer Find the value of x x 12 12 x 15 8

𝑎 2 + 𝑏 2 = 𝑐 2 250 2 + 350 2 = 𝑐 2 62,500+122,500= 𝑐 2 185,000= 𝑐 2 185,000 = 𝑐 2 185,000 =𝑐 𝑐=430.11626 𝑐≈430𝑚

Use the Pythagorean Thm. to find the height first! Finding the Area using the Pythagorean Theorem Use the Pythagorean Thm. to find the height first! 12 m 12 m 10 2 + ℎ 2 = 12 2 100+ ℎ 2 =144 ℎ 2 =44 ℎ 2 = 44 ℎ=2 11 20 m 𝐴=𝑏ℎ 𝐴= 1 2 (20)(2 11) 𝐴=20 11

Converse of the Pythagorean Theorem (Thm Converse of the Pythagorean Theorem (Thm. 7-5): If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

𝑎 2 + 𝑏 2 = 𝑐 2 4 2 + 6 2 = 7 2 16+36=49 52≠49 Since the sum of the square of the legs is not equal the square of the hypotenuse, this cannot be a right triangle.

𝑎 2 + 𝑏 2 = 𝑐 2 13 2 + 84 2 = 85 2 169+7,056=7,225 7,225=7,225 Since the sum of the square of the legs is equal to the square of the hypotenuse, this is a right triangle.

Theorem 7-6: Theorem 7-7:

Since the square of the hypotenuse The lengths of the sides of a triangle are given. Classify each triangle as acute, obtuse, or right. 𝑐 2 = 𝑎 2 + 𝑏 2 14 2 = 6 2 + 11 2 196=36+121 196=157 196>157 Since the square of the hypotenuse is greater than the sum of the square of the legs, the triangle is an obtuse triangle.

Since the square of the hypotenuse is less than the sum of the square The lengths of the sides of a triangle are given. Classify each triangle as acute, obtuse, or right. 𝑐 2 = 𝑎 2 + 𝑏 2 15 2 = 12 2 + 13 2 225=144+169 225=313 225<313 Since the square of the hypotenuse is less than the sum of the square of the legs, the triangle is an acute triangle.

Assignment Pg. 361 #’s 2 – 34 even