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CH 8 Right Triangles

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Geometric Mean of 2 #’s If you are given two numbers a and b you can find the geometric mean. a # = # b 3 x = x 27 Ex ) 3 and 27 X² = 81 √X² = √ 81 X = 9

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Lesson 7-1: Geometric Mean 3 Therefore ……….. To find the geometric mean between 7 and , ___, 28label the missing term x write a proportion cross multiply solve

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TOOL #1 Geometric Mean Geometric Mean – used with right triangles where all 3 altitudes are visible a d e m b

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Car Trick a d e m m d = e m

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Other Car Trick a d e m a d = d+e a

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Try it ! Lesson 7-1: Geometric Mean 7 Given: d = 4 and e = 10 Find: a = ___ b = ___ c = ___ f = ___

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Solution: Lesson 7-1: Geometric Mean 8 Proportions Answers

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TOOL #2 Pythagorean Theorem In a right triangle, the Leg²² + Leg² = Hypotenuse² a² + b² = c² 3 4 x 3² + 4² = x² = x² 25= x² √25=√ x² 5=x

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Converse of Right Triangle If the small side + small side = longest side, then the triangle is a ________________ If the small side + small side < longest side then the triangle is an _________________ If the small side + small side > longest side then the triangle is an __________________ RIGHT OBTUSE ACUTE

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Lesson 8-3 Lesson 7-3: Special Right Triangles 11 Special Right Triangles

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45°-45°-90° Special Right Triangle In a triangle 45°-45°-90°, the hypotenuse is times as long as a leg. Lesson 7-3: Special Right Triangles 12 45° Hypotenuse X X X Leg Example: 45° 5 cm 5 cm

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30°-60°-90° Special Right Triangle In a triangle 30°-60°-90°, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg. Lesson 7-3: Special Right Triangles 13 30° 60° Hypotenuse X 2X X Longer Leg Shorter Leg Example: 30° 60° 10 cm 5 cm

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Example: Find the value of a and b. Lesson 7-3: Special Right Triangles 14 60° 7 cm a b Step 1: Find the missing angle measure.30° Step 2: Decide which special right triangle applies.30°-60°-90° Step 3: Match the 30°-60°-90° pattern with the problem. 30° 60° x 2x a = cm b = 14 cm Step 5: Solve for a and b Step 4: From the pattern, we know that x = 7, b = 2x, and a = x.

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Example: Find the value of a and b. Lesson 7-3: Special Right Triangles 15 45° 7 cm a b Step 1: Find the missing angle measure.45° Step 2: Decide which special right triangle applies.45°-45°-90° Step 3: Match the 45°-45°-90° pattern with the problem. 45° x x x Step 4: From the pattern, we know that x = 7, a = x, and b = x. a = 7 cm b = 7 cm Step 5: Solve for a and b

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Example: Find the value of a and b. 12√3 Lesson 7-3: Special Right Triangles 16 60° 12 cm a b Step 1: Find the missing angle measure.30° Step 2: Decide which special right triangle applies.30°-60°-90° Step 3: Match the 30°-60°-90° pattern with the problem. 30° 60° x 2x a = cm b = 24 cm Step 5: Solve for a and b Step 4: From the pattern, we know that x = 12, b = 2x, and a = x.

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Example: Find the value of a and b. Lesson 7-3: Special Right Triangles 17 45° 9cm a b Step 1: Find the missing angle measure.45° Step 2: Decide which special right triangle applies.45°-45°-90° Step 3: Match the 45°-45°-90° pattern with the problem. 45° x x x Step 4: From the pattern, we know that x = 9, a = x, and b = x. a = 9 cm b = 9 cm Step 5: Solve for a and b

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