A b c. P ROVING THE P YTHAGOREAN T HEOREM THEOREM THEOREM 8-1 Pythagorean Theorem c 2 = a 2 + b 2 b a c In a right triangle, the square of the length.

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Presentation transcript:

a b c

P ROVING THE P YTHAGOREAN T HEOREM THEOREM THEOREM 8-1 Pythagorean Theorem c 2 = a 2 + b 2 b a c 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. (Converse is Theorem 8-2)

Math Review Perfect Squares –1–1–1–12 = 1 –2–2–2–22 = 4 –3–3–3–32 = 9 –4–4–4–42 = 16 –5–5–5–52 = 25 –6–6–6–62 = 36 –7–7–7–72 = 49 –8–8–8–82 = 64 –9–9–9–92 = 81 Radicals √3 √3 = √9 = 3 √40 = √4 10 = 2√10 √80 = √420 = 2√45 = 22√5 = 4√5

Example 2: Solve for x and Simplify the radical 20 x 8 a 2 + b 2 = c 2 x = 20 2 x = 400 x 2 = 336 √x 2 = √336 x = √(16)(21) x = 4√(21)

Example:

Example 3: Find the area of ΔDCE 12m 20m A = ½ bh = ½ (20m)(6.6m) = 66m 2 a 2 + b 2 = c b 2 = b 2 = 144 b 2 = 44 b = √44 b = √(4)(11) b = 2√(11) D C E b = 6.6m

Example 4: Are the following Δs, rt. Δ a 2 + b 2 = c = = = 7225 YESS ! a 2 + b 2 = c = = ≠ 2500 Nope!

Non-Right Δs Th(8-3) If the square of the length of the longest side of a Δ is greater than the sum of the squares of the lengths of the other 2 sides, the Δ is obtuse. If a 2 + b 2 < c 2, then the Δ is obtuse.

Th(8-4) If the square of the length of the longest side of a Δ is less than the sum of the squares of the lengths of the other 2 sides, then the Δ is acute. If a 2 + b 2 > c 2, then the Δ is acute.

Is the Δ Right, Obtuse, or Acute? Ex. 5 Sides of 6, 11, 14 a 2 + b 2 = c = = < 196 Obtuse Ex. 6 Sides of 15, 13, 12 a 2 + b 2 = c = = > 225 Acute

Example: