Exercise Find the hypotenuse in a right triangle

Exercise Find the missing side length in a right triangle with leg b =

45 -45 Right Triangle If each leg of a 45 -45 right triangle is

Example 1 Find the length of the hypotenuse in the 45 -45 right triangle.

Example 2 Find the length of the legs in the 45 -45 right triangle.

Example If a = 2, what is c? 45° c a b 2√ 2

Example If c = 8, what is a? 45° c a b 4√ 2

Example If the perimeter is 6 + 6 √ 2, what are a and

Example If the perimeter is 20, what is a? 45° c a b ≈

Example Are 5, 5, and 7 the sides of a 45 -45 right triangle?

30 -60 Right Triangle If the short leg of a 30 -60 right triangle

Example 3 Find the missing lengths x and y in the triangle. 60° x

Example 4 Find the short leg and hypotenuse of a 30 -60 right Δ

Example 5 Find the long leg and short leg of a 30 -60 right

Example If a = 1, what are b and c? 60° c a b

Example If c = 7, what is a? 60° c a b 7 =

Example If a = 3, what is the perimeter? 60° c a b 9

Example If the perimeter is 25, what is c? 60° c a b 10.

Example We know that 3, 4, and 5 are the lengths of the sides

Exercise Find the other two sides in a 30 -60 right triangle with a

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Exercise Find the hypotenuse in a right triangle with legs a = 3 and b = 4. 5

Exercise Find the hypotenuse in a right triangle with legs a = 3 and b = 3. 3√ 2

Exercise Find the missing side length in a right triangle with leg b = 4 and hypotenuse 4 √ 2. 4

Exercise Find the hypotenuse in a right triangle with legs a = 3 and b = 3 √ 3. 6

Exercise Find the missing side length in a right triangle with leg b = 5 √ 3 and hypotenuse 10. 5

A C B

45° 1 c 45° 1

45 -45 Right Triangle If each leg of a 45 -45 right triangle is a units long, then the hypotenuse is a √ 2 units long. 45° a√ 2 a a 45°

Example 1 Find the length of the hypotenuse in the 45 -45 right triangle. 45° x 3 3 3√ 2 45°

Example 2 Find the length of the legs in the 45 -45 right triangle. 45° 5√ 2 a a 5 45°

Example If a = 2, what is c? 45° c a b 2√ 2 45°

Example If c = 8, what is a? 45° c a b 4√ 2 45°

Example If the perimeter is 6 + 6 √ 2, what are a and c? 45° c a b 45° a = 3 √ 2 and c = 6

Example If the perimeter is 20, what is a? 45° c a b ≈ 5. 86 45°

Example Are 5, 5, and 7 the sides of a 45 -45 right triangle? no

60° 30° 60°

30 -60 Right Triangle If the short leg of a 30 -60 right triangle is a units long, then the long leg is a √ 3 units long and the hypotenuse is 2 a units long. 60° a 2 a a√ 3 30°

Example 3 Find the missing lengths x and y in the triangle. 60° x 30° y 4 Since a = 4, the long leg is 4 √ 3 and the hypotenuse is 2(4) = 8 units.

Example 4 Find the short leg and hypotenuse of a 30 -60 right Δ whose long leg is 6 √ 3. Since the long leg of a 30 -60 right triangle is a √ 3 = 6 √ 3, it follows that the short leg it a = 6. Then the hypotenuse is 2 a = 2(6) = 12.

Example 5 Find the long leg and short leg of a 30 -60 right triangle whose hypotenuse is 9. The hypotenuse is 2 a = 9, so the short leg is a = 4. 5. The long leg is a √ 3 = 4. 5 √ 3.

Example If a = 1, what are b and c? 60° c a b 30° b = √ 3 and c = 2

Example If c = 7, what is a? 60° c a b 7 = 3. 5 2 30°

Example If a = 3, what is the perimeter? 60° c a b 9 + 3 √ 3 ≈ 14. 2 30°

Example If the perimeter is 25, what is c? 60° c a b 10. 566 30°

Example We know that 3, 4, and 5 are the lengths of the sides of a right triangle. Are they the sides of a 30 -60 right triangle? no; 5 ≠ 2(3)

Exercise Find the other two sides in a 30 -60 right triangle with a short leg of √ 5. long leg: √ 15; hypotenuse: 2 √ 5 sides: 2. 2, 3. 9, 4. 5

Exercise Find the other two sides in a 30 -60 right triangle with a short leg of 2 √ 7. long leg: 2 √ 21; hypotenuse: 4 √ 7 sides: 5. 3, 9. 2, 10. 6

Exercise Find the other two sides in a 30 -60 right triangle with a long leg of 20 √ 3. short leg: 20; hypotenuse: 40 sides: 20, 34. 6, 40