1. Mrs. McConaughyGeometry1 Lesson 7.3 Two Special Right Triangles Objectives:  To use properties of 45-45-90 triangles  To use properties of 30-60-90 triangles

2. Mrs. McConaughyGeometry2 Isosceles Right Triangle Theorem ISOSCELES RIGHT TRIANGLE THEOREM: In an isosceles right triangle, if the legs have length, l, then the hypotenuse has length ____. l √2 NOTE: If you are given the length of the hypotenuse, you can determine the length of a side by dividing it by_________________________ ___________________________. √2, then rationalizing the denominator, when necessary.

3. Mrs. McConaughyGeometry3 EXAMPLES: Find the length of the hypotenuse in each isosceles triangle below. 3 √2 4 √2 5 √2 6 √2 7 √2 12 √2

4. Mrs. McConaughyGeometry4 Recall: Triangle Inequalities If two angles of a triangle are not congruent, then the longest side lies opposite the _______ angle and the shortest side lies opposite the ________ angle. smallest largest

5. Mrs. McConaughyGeometry5 30-60-90 TRIANGLE THEOREM 30-60-90 TRIANGLE THEOREM: In a 30-60- 90 triangle, if the side opposite the 30 degree angle has length, l, the hypotenuse has length _______. NOTE: These triangles are sometimes referred to as 1-2-√3 right triangles. 2l2l

6. Mrs. McConaughyGeometry6 Easy way to remember the relationship among angles and sides in 30-60-90 triangles: 1. Rank order the following numbers from smallest to largest: 1, 2, √3 2. Now, use the Triangle Inequality Theorem to place the side lengths 1 l, √3 l, 2 l opposite the appropriate angles in a 30-60-90 triangle. 1, √3, 2 30 60 1l1l 2l2l l √3 NOTE: It is usually easier to determine the length of the shortest and longest sides, initially.

7. Mrs. McConaughyGeometry7 Find the length of each indicated side: 30 60 ____ NOTE: The length of one side will be provided by your instructor.

8. Mrs. McConaughyGeometry8 Find the length of each indicated side.

9. Mrs. McConaughyGeometry9 c a b Pythagorean Theorem c 2 = a 2 + b 2 In summary: We can find the lengths of sides in right triangles by using: 30-60-90 ∆ 30 2l l √3 60 l 45-45-90 ∆ l 45 l l √2 45 Pythagorean Primitives 3 4 5 5 12 13 8 15 17 7 24 25 …and their multiples !

10. Mrs. McConaughyGeometry10 Putting it all together: Find the length of each indicated side. 8 ∙ 3 8 ∙ 4 8 ∙ __ = __ 20 20 √3 5 40

11. Mrs. McConaughyGeometry11 Homework Assignment: Special Right Triangles WS (1-10 all, 12)