Bellringer Solve for y. 1. y=5√5 2.y=7 The lengths of the sides of a triangle are given. Classify each triangle as acute, right, or obtuse. 3. 3, 8, 10 obtuse 4. 4, 5, 7 5. 12, 15, 19 acute Solve for y. 1. y=5√5 2.y=7

Sect.8-2 Special Right Triangles Geometry: Chapter 8 Right Triangles and Trigonometry

Review Review Essential Understanding We learned how the Pythagorean theorem helped us find the third side of a right triangle. Also, we examined the converse of the Pythagorean Theorem to determine whether it is right, obtuse, or acute triangle.

Lesson’s Purpose What are the properties of special right triangles? Essential Question Objective What are the properties of special right triangles? A 45-45-90 triangle is an isosceles triangle. A 30-60-90 triangle has a hypotenuse that is twice as longer as the shorter leg. In this section we are going to learn about how all similar right triangles have constant ratios of side lengths of special right triangles. This will provide a bridge between the study of Pythagorean Theorem and Trigonometry.

Theorem 8-5 45˚-45˚-90˚ Triangle Theorem The lengths of the sides of a 45°- 45°- 90° triangle are in the ratio of 1 :1 :√2.

Example #1 Solution: Step 1: This is a right triangle with two equal sides so it must be a 45°- 45°- 90° triangle. Step 2: You are given that the both the sides are 3. If the first and second value of the ratio n:n:n√2 is 3 then the length of the third side is 3√2 Answer: The length of the hypotenuse 3√2is  inches. Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are both 3 inches. 3 3

Helpful Hint You can also recognize a 45°- 45°- 90° triangle by the angles. As long as you know that one of the angles in the right-angle triangle is 45° then it must be a 45°- 45°- 90° special right triangle.

Example #2 Solution: Step 1: This is a right triangle with a 45°so it must be a 45°- 45°- 90° triangle. You are given that the hypotenuse is 4√2 . If the third value of the ratio n:n:n√2 is 4√2 then the lengths of the other two sides must 4. Answer: The lengths of the two sides are both 4 inches.   Find the lengths of the other two sides of a right triangle if the length of the hypotenuse 4√2 is inches and one of the angles is 45°. 45 4√2

Theorem 8-6 30˚-60˚-90˚ Triangle Theorem his is right triangle whose angles are 30°, 60°and 90°. The lengths of the sides of a 30°- 60°- 90° triangle are in the ratio of 1 :√3 :2 .

Example #3 Solution: Step 1: Test the ratio of the lengths to see if it fits the  ratio. n:n√3:2n       4: 4 √3:?=n:n√3:2n       Step 2:  Yes, it is a 30°- 60°- 90° triangle for n = 4 Step 3:  Calculate the third side. 2n = 2×4 = 8 Answer: The length of the hypotenuse is 8 inches. Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are 4 inches and 4√3 inches. 4 4√3

Helpful Hint You can also recognize a 30°- 60°- 90° triangle by the angles. As long as you know that one of the angles in the right-angle triangle is either 30° or 60° then it must be a 30°- 60°- 90° special right triangle.

Example #4 Solution: Step 1: This is a right triangle with a 30° angle so it must be a 30°- 60°- 90° triangle. You are given that the hypotenuse is 8. Substituting 8 into the third value of the ratio n:n√3:2n  , we get that 2n = 8 Þ n = 4. Substituting n = 4 into the first and second value of the ratio we get that the other two sides are 4 and 4 √3. Answer: The lengths of the two sides are 4 inches and 4 √3 inches. Find the lengths of the other two sides of a right triangle if the length of the hypotenuse is 8 inches and one of the angles is 30°. 8 30

A 3-4-5 Triangle  Is right triangle whose lengths are in the ratio of 3:4:5. When you are given the lengths of two sides of a right triangle, check the ratio of the lengths to see if it fits the 3:4:5 ratio. Side1 : Side2 : Hypotenuse = 3n : 4n : 5n

Example #5: Find the length of the hypotenuse of a right triangle if the lengths of the other two sides are 6 inches and 8 inches. Solution: Step 1: Test the ratio of the lengths to see if it fits the 3n : 4n : 5n ratio.               6 : 8 : ? = 3(2) : 4(2) : ? Step 2:  Yes, it is a 3-4-5 triangle for n = 2. Step 3: Calculate the third side 5n = 5×2 = 10 The length of the hypotenuse is 10 inches. 8 6

Example #6 Find the length of one side of a right triangle if the length of the hypotenuse is 15 inches and the length of the other side is 12 inches. Solution: Step 1: Test the ratio of the lengths to see if it fits the 3n : 4n : 5n ratio.               ? : 12 : 15 = ? : 4(3) : 5(3) Step 2: Yes, it is a 3-4-5 triangle for n = 3. Step 3: Calculate the third side 3n = 3×3 = 9 The length of the side is 9 inches. 15 12

Real World Connections

Summary Essential Understanding A 45˚-45˚-90˚ triangle is an isosceles triangle. The legs are congruent and the hypotenuse length is √2 times the length of a leg. A 30˚-60˚-90˚ triangle has a hypotenuse that is twice as long as the shorter leg and the longer leg that is √3 times the shorter leg.

Ticket out and Homework What are special right triangles? Why are they studied? 30-60-90 and 45-45-90 triangles. They are studied because of their special properties and can be used as a shortcut to find missing sides of these triangles. Homework: Pg 528-30 #’s 8-22 even only