Warm-Up! Find the length of the missing side. Write your answer in simplest radical form. 1.) 4 x 𝒙=𝟒 𝟐 5 5 2 x 2.) 𝒙=𝟓 𝟑 3.) A triangle has side lengths of 9 in, 10 in, and 12 in. Classify the triangle by it’s angles. (HINT: Acute, Obtuse, or Right?) The triangle would be an acute triangle because 144 < 181. 4.) Joseph has an 8 foot ladder. The ladder reaches up 7 feet on the wall. In order for the ladder to be stable, the base must be less than 4 feet away from the wall. Is Joseph’s ladder stable? Round your answer to the nearest tenth and explain your answer. (HINT: Draw a picture!) The base of the ladder would be 3.9 feet away so Joseph’s ladder would be safe.

Special Right Triangles Mr. Riddle

45-45-90 Triangles In a 45-45-90 triangle, both legs are congruent and the length of the hypotenuse is 2 times the length of the leg. Hypotenuse = 2 ∙𝐿𝑒𝑔 x 𝑥 2 45°

Example 1 – Finding the Length of the Hypotenuse Find the value of each variable. Leave your answer in simplest radical form. a.) b.) 45° x 2 2 9 45° h ℎ= 2 ∙9 𝒉=𝟗 𝟐 x=2 2 ∙ 2 𝑥=2 4 𝑥=2∙2 𝒙=𝟒

YOU TRY! Find the length of the hypotenuse of a 45- 45-90 triangle with legs of length 5 3 . ℎ=5 3 ∙ 2 𝒉=𝟓 𝟔

Example 2 – Finding the Length of a Leg What is the value of x? 𝑥∙ 2 =6 𝑥= 6 2 𝑥= 6 2 ∙ 2 2 = 6 2 2 𝒙=𝟑 𝟐 45° x 6

You Try! Find the length of a leg of a 45-45-90 triangle with a hypotenuse of length 10. 𝑥∙ 2 =10 𝑥= 10 2 𝑥= 10 2 ∙ 2 2 = 10 2 2 𝒙=𝟓 𝟐

Example 3 – REAL APPLICATION A high school baseball field is a square. The distance from base to base is 90 feet. To the nearest foot, how far does a catcher throw a ball from home plate to second base. Since the triangle formed is both isosceles and a right triangle, it must be a 45-45-90 triangle. From home to second base would represent the hypotenuse of that triangle. 90 ft 𝒉=𝟗𝟎∗ 𝟐 𝒉=𝟏𝟐𝟕.𝟐𝟕𝟗𝟐𝟐𝟎𝟔 𝒉≈𝟏𝟐𝟕 𝒇𝒆𝒆𝒕

You try! A Softball field has 60 foot base paths. How far does the catcher have to throw from home to second base? 60 ft 𝒉=𝟔𝟎∗ 𝟐 𝒉=𝟖𝟒.𝟖𝟓𝟐𝟖𝟏𝟒 𝒉≈𝟖𝟓 𝒇𝒆𝒆𝒕

30-60-90 Triangles 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 x 𝑥 3 2x 60° 30°

Example 4 – Using the length of one side Find the value of each variable. 5=𝑑 3 𝑑= 5 3 ∙ 3 3 = 𝟓 𝟑 𝟑 𝑓=2𝑑 𝑓=2 5 3 3 = 𝟏𝟎 𝟑 𝟑 d 5 f 60° 30°

You Try! Find the value of each variable. 𝑥= 8 2 =𝟒 𝑦=𝑥 3 =𝟒 𝟑 8 x 𝑦 𝑦=𝑥 3 =𝟒 𝟑 x 𝑦 8 60° 30°

Homework ALEKS: Assignment # 40