Geometry 9.2 Special Right Triangles 9.2 Day 1 Warm Up Find the value of x. Give answers as simplified radicals. 1. 2. 3. 4. 10 x x 3 x February 27, 2019 Geometry 9.2 Special Right Triangles

9.2 Special Right Triangles Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles 9.2 Essential Question What is the relationship among the side lengths of 45°- 45°- 90° triangles? 30°- 60°- 90° triangles? February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Goals Know the side lengths of special right triangles. Use special right triangles to solve problems. February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.4 Special Right Triangles First, radical review: A radical in simplest form has no perfect squares in the radicand. February 27, 2019 Geometry 9.4 Special Right Triangles

Rationalizing Denominators This means no square roots allowed in the denominator of fractions. General rule is to multiply numerator and denominator by the radical and simplify. February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Examples February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles February 27, 2019 Geometry 9.2 Special Right Triangles

Constructing Special Triangles Given a Square. Draw one diagonal. The diagonal bisects the angles. 1 & 2 measure ______. 2 1 45 February 27, 2019 Geometry 9.2 Special Right Triangles

Constructing Special Triangles Clean it up – keep only the triangle on the right. 45 45 February 27, 2019 Geometry 9.2 Special Right Triangles

Constructing Special Triangles In the original square, each side was the same. Label them a. 45 a 45 a February 27, 2019 Geometry 9.2 Special Right Triangles

Constructing Special Triangles Solve for the hypotenuse, c. 45 𝑎 2 c a 45 a February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles 45-45-90 Triangle Theorem (Thm 9.4) In a 45°- 45°- 90° triangle, the hypotenuse is 2 times as long as each leg. 45 a 𝑎 2 February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Example 1 Solve for x & y. 45 45 a a2 y 5 x 52 a 45 5 February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Example 2 Solve for x & y. 45 45 a a2 x 82 8 a 45 8 y February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Example 3: A tougher one. Solve for x & y. 45 45 a a2 10 x a 45 y February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Example 3 Solution 45 10 x 52 45 y 52 February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Learn the pattern! 45 a a2 February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Try it. Find x and y. 45 8 y 42 45 x 42 February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Shortcut pattern: 45 2a 𝑎 2 45 𝑎 2 February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Example 4 Solve for x & y 40 y 202 x 202 February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Example 5 A designer wants to put a rope light on the diagonal of a square dance floor. If the floor measures 30 ft. on a side, how long does the rope light need to be? February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Example 5 Solution 302  42.43 ft.  42 ft 5 in. 45 302 30 ft. 45 30 ft. February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles 9.2 Day 2 Warm Up Solve for the variable. 1. 2. 3. 4. X 12 4 3 8 X 4 February 27, 2019 Geometry 9.2 Special Right Triangles

Constructing Special Triangles Construct an equilateral triangle. Each angle measures ______. 60 60 60 60 February 27, 2019 Geometry 9.2 Special Right Triangles

Constructing Special Triangles Draw an altitude. It is perpendicular to the base. It also bisects the vertex angle. The altitude divides the triangle into two congruent triangles with angles measures of 30 - 60 - 90. 60 30 60 60 February 27, 2019 Geometry 9.2 Special Right Triangles

Constructing Special Triangles Clean up the drawing – only keep the triangle on the left. 30 60 60 February 27, 2019 Geometry 9.2 Special Right Triangles

Constructing Special Triangles This is called a 30 – 60 – 90 triangle. 30 60 February 27, 2019 Geometry 9.2 Special Right Triangles

Constructing Special Triangles Give the original side an arbitrary length. Call it 2a. 30 2a 60 February 27, 2019 Geometry 9.2 Special Right Triangles

Constructing Special Triangles What is the length of the base? 30 2a 60 a ? 2a February 27, 2019 Geometry 9.2 Special Right Triangles

Constructing Special Triangles Now find the height of the triangle, h. a2 + h2 = (2a)2 30 2a h 60 a February 27, 2019 Geometry 9.2 Special Right Triangles

Constructing Special Triangles 30 2a h 60 a February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles 60 30 2a a Short leg Long leg Hypotenuse 30-60-90 Triangle Theorem (Theorem 9.5) In a 30°- 60°- 90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is 3 times as long as the shorter leg. February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles 60 30 60 30 2a 3 a 3 60 30 2a a 3𝑎 2 𝑎 3 2 3a a 3 February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Example 6 Find x & y. 30 60 30 2a a 12 y 63 60 x 6 February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Example 7 Find x & y. 30 60 30 2a a 16 83 x y 60 8 February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Example 8 Find x & y. 30 60 30 2a a y 20 3 40 60 x 20 February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Example 9 Find x & y. 30 y 12 60 x February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Example 9 Solution 30 y 12 60 x February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Learn the pattern! 30 2a 60 a February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Try it. Find x and y. 30 y x 153 30 60 15 February 27, 2019 Geometry 9.2 Special Right Triangles

Another one: Find h and y. 30 h 73 14 60 y 7 February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Again: Find x and y. 30 x 4 2 𝟑 60 y 2 February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Summary 60 30 2a a 45 a a 2 45-45-90 30-60-90 In Trigonometry, you will understand why these triangle are so important. February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Quick Quiz – 5 problems 30 x y 12 1. Find x & y. 123 24 February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles 2. Find x & y. 45 7 2 y x February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles 3. Find x & y. 60 50 x 25 y 253 February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles 4. Find x & y. 62 x 45 y February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles 5. Find x & y. 75 60 x y February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles 6. Find x & y. 60 y 9 3 x February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles One More Time… 60 30 2a a 45 a a2 February 27, 2019 Geometry 9.2 Special Right Triangles

Geometry 9.2 Special Right Triangles Homework February 27, 2019 Geometry 9.2 Special Right Triangles