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Assignment P. 461-464: 1-18, 23-25, 28, 30, 31, 34, 36 Challenge Problems

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Warm-Up Solve the quadratic equation:

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7.4: Special Right Triangles Objectives: 1.To use the properties of 45-45-90 and 30-60-90 right triangles to solve problems

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Investigation 1 This triangle is also referred to as a 45-45-90 right triangle because each of its acute angles measures 45°. Folding a square in half can make one of these triangles. In this investigation, you will discover a relationship between the lengths of the legs and the hypotenuse of an isosceles right triangle.

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Investigation 1 Find the length of the hypotenuse of each isosceles right triangle. Simplify the square root each time to reveal a pattern.

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Investigation 1 Did you notice something interesting about the relationship between the length of the hypotenuse and the length of the legs in each problem of this investigation?

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Special Right Triangle Theorem 45°-45°-90° Triangle Theorem In a 45°-45°-90° triangle, the hypotenuse is times as long as each leg.

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Example 1 Use deductive reasoning to verify the Isosceles Right Triangle Conjecture.

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Example 2 A fence around a square garden has a perimeter of 48 feet. Find the approximate length of the diagonal of this square garden.

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FoxTrot

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Investigation 2 The second special right triangle is the 30- 60-90 right triangle, which is half of an equilateral triangle. Let’s start by using a little deductive reasoning to reveal a useful relationship in 30-60- 90 right triangles.

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Investigation 3 Triangle ABC is equilateral, and segment CD is an altitude. 1.What are m<A and m<B? 2.What are m<ADC and m<BDC? 3.What are m<ACD and m<BCD? 4.Is Δ ADC = Δ BDC? Why? 5.Is AD=BD? Why? ~

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Investigation 2 Notice that altitude CD divides the equilateral triangle into two right triangles with acute angles that measure 30° and 60°. Look at just one of the 30-60-90 right triangles. How do AC and AD compare?Conjecture: In a 30°-60°-90° right triangle, if the side opposite the 30° angle has length x, then the hypotenuse has length -?-.

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Investigation 2 Find the length of the indicated side in each right triangle by using the conjecture you just made.

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Investigation 2 Now use the previous conjecture and the Pythagorean formula to find the length of each indicated side.

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Investigation 2 You should have notice a pattern in your answers. Combine your observations with you latest conjecture and state your next conjecture.

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Special Right Triangle Theorem 30°-60°-90° Triangle Theorem In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg.

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Two Special Right Triangles

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Example 3 Find the value of each variable. Write your answer in simplest radical form. 1. 2. 3. X=4 X=13 Y=26 Y=12

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Example 4 Find the value of each variable. Write your answer in simplest radical form. 1. 2. 3.

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Example 5 What is the area of an equilateral triangle with a side length of 4 cm? 4 cm

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Example 6: SAT In the figure, what is the ratio of RW to WS? X = 2

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Assignment P. 461-464: 1-18, 23-25, 28, 30, 31, 34, 36 Challenge Problems

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