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8.2 Special Right Triangles Geometry Mr. Jacob P. Gray Franklin County High School Click for next Slide 8.2 Special Right Triangles Objectives • • To use the properties of 45˚-45˚-90˚ triangles To use the properties of 30˚-60˚-90˚ triangles Click for next Slide Take notes on the following: Essential Understanding Certain right triangles have properties that allow you to use shortcuts to determine side lengths without using the Pythagorean Theorem. Right Isosceles Triangle The acute angles of an Isosceles Right Triangle are both 45˚. Another name for this is a 45˚-45˚-90˚ triangle. x x 45˚ 45˚ y 𝒙𝟐 + 𝒙𝟐 = 𝒚𝟐 Use the Pythagorean Theorem 𝟐𝒙𝟐 = 𝒚𝟐 Simplify 𝒙√𝟐 = 𝒚 Take the positive square root of each side Click for next Slide Take notes on the following: Theorem 8-5 45˚-45˚-90˚ Triangle Theorem In a 45˚-45˚-90˚ triangle, both legs are congruent and the length of the hypotenuse is √2 times the length of a leg. s√2 Hypotenuse = √2 · leg 45˚ s 45˚ s Click for next Slide Here is a problem to practice. Problem 1) Finding the Length of the Hypotenuse What is the value of each variable? 1 2 9 45˚ x 45˚ 45˚ h Hypotenuse = √2 · leg 45˚ 2√2 Click for next Slide Here is another problem to practice. Problem 2) Finding the Length of a Leg What is the value of x? Hypotenuse = √2 · leg 6 45˚ 45˚ x Click for next Slide Here is another problem to practice. Problem 3) Finding Distance Softball A softball diamond is a square. The distance from base to base is 60 ft. To the nearest foot, how far does the catcher throw the ball from home to second base? Hypotenuse = √2 · leg 60 ft d 𝒅 = 𝟔𝟎 · √𝟐 Use a calculator. 𝒅 ≈ 𝟖𝟒. 𝟖𝟓𝟐𝟖… The catcher throws the ball about 85 ft from home plate to second base. Click for next Slide Another type of special right triangle is a 30˚-60˚-90˚ triangle. Theorem 8-6 30˚-60˚-90˚ Triangle Theorem In a 30˚-60˚-90˚ triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is √3 times the length of the shorter leg. hypotenuse = 2 · shorter leg longer leg = √3 · shorter leg Click for next Slide 30˚ 2s 60˚ s s√3 Here is a problem to practice. Problem 4) Using the Length of One Side Algebra What is the value of d in simplest radical form? 5 d 60˚ 30˚ f 𝒍𝒐𝒏𝒈𝒆𝒓 𝒍𝒆𝒈 = √𝟑 · 𝒔𝒉𝒐𝒓𝒕𝒆𝒓 𝒍𝒆𝒈 𝟓 = 𝒅√𝟑 𝟓 𝟑 = 𝒅 = 𝒅 𝟑 𝟑 𝟓 𝟑 ∙ √𝟑 𝟓√𝟑 = 𝟑 𝟑 Click for next Slide Here is the last problem to practice. Problem 5) Applying the 30˚-60˚-90˚ Triangle Theorem Jewelry Making An artisan makes pendants in the shape of equilateral triangles. The height of each pendant is 18mm. What is the length s of each side of a pendant to the nearest tenth of a millimeter? 𝑙𝑜𝑛𝑔𝑒𝑟 𝑙𝑒𝑔 = 3 · 𝑠ℎ𝑜𝑟𝑡𝑒𝑟 𝑙𝑒𝑔 18 = √3 1 𝑠 2 S 18mm 3 18 = 𝑠 2 2 3 ∙ 18 = 𝑠 𝑠 ≈ 20.7846. . Each side is about 20.8 mm long. Click for next Slide Classwork Pages 503-505 7-21 all 23-29 all 34, 35, 37, 38, 39 STOP