Pearson Unit 3 Topic 10: Right Triangles and Trigonometry 10-2: Special Right Triangles Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

TEKS Focus: (9)(B) Apply the relationships in special right triangles 30-60-90 and 45-45-90 and the Pythagorean Theorem, including Pythagorean triples, to solve problems. (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. (1)(A) Apply mathematics to problems arising in everyday life, society, and the workplace. (6)(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems.

DISCOVERY 1: Find the measure of the hypotenuse in simplest radical form. What did you notice? 17 9 3 3 9 17 3 17 9 9 2 + 9 2 = 𝑐 2 81+81= 𝑐 2 162= 𝑐 2 162 = 𝑐 81 2 =𝑐 9 2 =𝑐 3 2 + 3= 𝑐 2 9+9= 𝑐 2 18= 𝑐 2 18 = 𝑐 9 2 =𝑐 3 2 =𝑐 17 2 + 17 2 = 𝑐 2 289+289= 𝑐 2 578= 𝑐 2 578 = 𝑐 289 2 =𝑐 17 2 =𝑐 The hypotenuse is the leg times the square root of 2.

The long leg of the 30-60-90 triangle DISCOVERY 2: Find the measure of the altitude of these equilateral triangles in simplest radical form. What did you notice? 8 2 + 𝑏 2 = 16 2 64+ 𝑏 2 = 256 𝑏 2 = 192 𝑏= 192 𝑏= 64 3 𝑏=8 3 5 2 + 𝑏 2 = 10 2 25+ 𝑏 2 = 100 𝑏 2 = 75 𝑏=75 𝑏= 25 3 𝑏=5 3 10 8 16 5 60 60 5 8 The long leg of the 30-60-90 triangle Is the short leg times the , and the hypotenuse is twice the short leg.

Hypotenuse 45-45-90 (x, x, x ) X Leg

30-60-90 (x, x , 2x) 2x X Long Leg Short Leg Hypotenuse 30 60

A 45°-45°-90° triangle is one type of special right triangle. Another name for an isosceles right triangle is a 45°-45°-90° triangle. A 45°-45°-90° triangle is one type of special right triangle.

A 30°-60°-90° triangle is another special right triangle A 30°-60°-90° triangle is another special right triangle. You can use an equilateral triangle to find a relationship between its side lengths.

Example in Flip Book: Both legs = 4 6 . To find the hypotenuse: 4 6 45° To find the hypotenuse: 4 6 ∗ 2 4 12 4 4 ∗ 3 4∗2 3 8 3

Example in Flip Book: Both legs = 17. To find the hypotenuse: 17∗ 2 45° Both legs = 17. To find the hypotenuse: 17∗ 2 17 2

Example in Flip Book: 45° To find both legs: 8 3 8 3 2 8 3 2 ∗ 2 2 8 6 2 4 6

Example in Flip Book: 10 2 10 2 ∗ 2 2 10 2 2 5 2 10 To find both legs: 10 2 10 45° 10 2 ∗ 2 2 10 2 2 5 2

Example in Flip Book: To find both legs: 45° 3 2 3 2 2 = 3

Example in Flip Book: Will a 29” square TV fit in a cabinet that is 2 feet by 2 feet? TVs and computer screens are measured on the diagonal. 29 2 29 2 ∗ 2 2 29 2 2 𝑖𝑠 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑙𝑦 20.5 inches, 𝑠𝑜 𝑡ℎ𝑒 𝑇𝑉 𝑤𝑖𝑙𝑙 𝑓𝑖𝑡 𝑖𝑛 2 𝑓𝑡 𝑏𝑦 2 𝑓𝑡.

Example in Flip Book: To find the short leg: 18 2 =9 30° 60° 18 18 2 =9 Now find the long leg: 9∗ 3 =9 3

Example in Flip Book: To find the short leg: 17 3 3 =17 30° 60° 17 3 17 3 3 =17 Now find the hypotenuse: 17∗2=34

Example in Flip Book: To find the long leg: 15∗ 3 =15 3 30° 60° 15 15∗ 3 =15 3 Now find the hypotenuse: 15∗2=30

Example in Flip Book: To find the short leg: 10 6 2 =5 6 30° 60° 10 6 10 6 2 =5 6 Now find the long leg: 5 6 ∗ 3 =5 18 =5 9 2 =15 2

Example in Flip Book: To find the short leg: 8 3 2 =4 3 30° 60° 8 3 8 3 2 =4 3 Now find the long leg: 4 3 ∗ 3 =4 9 =4∗3 =12

Example in Flip Book: The perimeter of an equilateral triangle = 36 cm. Find the altitude (the height). Then find the area. The altitude of an equilateral triangle divides the triangle into two 30-60-90 triangles. Each side = 12 cm, so the short leg = 6 cm and the hypotenuse = 12 cm. The altitude (the long leg) is 6 3 . Area = ½ (base)(height) Area = ½ (12)(6 3 ) Area = 36 3 𝑐𝑚 2

The following examples are not in your Flip Book, but are good examples for you to read through.

Example: 1 Find the value of x. Give your answer in simplest radical form. By the Triangle Sum Theorem, the measure of the third angle in the triangle is 45°. So it is a 45°-45°-90° triangle with a leg length of 8.

Example: 2 Find the value of x. Give your answer in simplest radical form. The triangle is an isosceles right triangle, which is a 45°-45°-90° triangle. The length of the hypotenuse is 5. Divide both sides by 2 . Rationalize the denominator. Multiply 5 2 ∙ 2 2 Simplify

Example: 3 x = 20 Find the value of x. Give your answer in simplest radical form. By the Triangle Sum Theorem, the measure of the third angle in the triangle is 45°. So it is a 45°-45°-90° triangle with a leg length of x = 20 Simplify.

Example: 4 Find the value of x. Give your answer in simplest radical form. The triangle is an isosceles right triangle, which is a 45°-45°-90° triangle. The length of the hypotenuse is 16. Divide both sides by 2 . Rationalize the denominator. Multiply 16 2 ∙ 2 2 Simplify

Example: 5 Jana is cutting a square of material for a tablecloth. The table’s diagonal is 36 inches. She wants the diagonal of the tablecloth to be an extra 10 inches so it will hang over the edges of the table. What size square should Jana cut to make the tablecloth? Round to the nearest inch. Jana needs a 45°-45°-90° triangle with a hypotenuse of 36 + 10 = 46 inches. Picture Source: www.allaboutcleaners.com

Example: 6 59 – 16 = 43. Subtract the extra 16 cm Tessa wants to make a bandana for her dog by folding a square cloth with a side length of 42 cm into a 45-45-90 triangle. The folded bandana needs to be an extra 16 cm long so Tessa can tie it around the dogs neck. What would you expect the circumference of her dog’s neck to be? Round to the nearest centimeter. 59 – 16 = 43. Subtract the extra 16 cm The circumference of dog’s neck is about 43 cm.

Example: 7 Find the values of x and y. Give your answers in simplest radical form. 22 = 2x Hypotenuse = 2(shorter leg) 11 = x Divide both sides by 2. Substitute 11 for x.

Example: 8 Find the values of x and y. Give your answers in simplest radical form. Divide both sides by 3 . Rationalize the denominator. Multiply 15 3 ∙ 3 3 Simplify. y = 2x Hypotenuse = 2(shorter leg). Simplify.

Example: 9 Find the values of x and y. Give your answers in simplest radical form. Hypotenuse = 2(shorter leg) Divide both sides by 2. y = 27 Substitute for x.

Example: 10 Find the values of x and y. Give your answers in simplest radical form. y = 2(5) y = 10 Simplify.

Example: 11 Find the values of x and y. Give your answers in simplest radical form. 24 = 2x Hypotenuse = 2(shorter leg) 12 = x Divide both sides by 2. Substitute 12 for x.

Example: 12 Find the values of x and y. Give your answers in simplest radical form. Multiply 9 3 ∙ 3 3 Rationalize the denominator. Simplify. x = 2y Hypotenuse = 2(shorter leg) Simplify.

Example: 13 A B An ornamental pin is in the shape of an equilateral triangle. The length of each side is 6 centimeters. Josh will attach the fastener to the back along AB. Will the fastener fit if it is 4 centimeters long? Step 1 The equilateral triangle is divided into two 30°-60°-90° triangles. The height of the triangle is the length of the longer leg.

Example: 13 continued Step 2 Find the length x of the shorter leg. B Step 2 Find the length x of the shorter leg. 6 = 2x Hypotenuse = 2(shorter leg) 3 = x Divide both sides by 2. Step 3 Find the length h of the longer leg. The pin is approximately 5.2 centimeters high. So the fastener will fit because 4 cm < 5.2 cm.

Example: 14 What if a manufacturer wants to make a clock in the shape of an equilateral triangle with a height of 30 centimeters. What is the length of each side of the clock frame? Round to the nearest tenth. Step 1 The equilateral triangle is divided into two 30º-60º-90º triangles. The height of the triangle is the length of the longer leg.

Example: 14 continued Step 2 Find the length x of the shorter leg. Multiply 30 3 ∙ 3 3 Rationalize the denominator. Simplify. Step 3 Find the length y of the longer leg. y = 2x Hypotenuse = 2(shorter leg) Simplify. Each side is approximately 34.6 cm.

Example: 15 x3 2x x X3 = 18 3 3 X = 18 3 3 X = 6 3 X ≈ 10.4 mm