Find the hypotenuse in a right triangle with legs a = 3 and b = 4. 55 Exercise.

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

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

55

ExerciseExercise

ExerciseExerciseFind the hypotenuse in a right triangle with legs a = 3 and b = 3.

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

3 √ 23 √ 2

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

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

44

ExerciseExercise

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

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

66

ExerciseExercise

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

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

55

ExerciseExercise

AA

CC BB

45°45°

11cc

45°45°

11

a1a1

c√ 2c

√ 2==

45-45 Right Triangle45-45 Right Triangle

If each leg of a 45-45 right triangle is a units long, then the

hypotenuse is a √ 2 units long.

If each leg of a 45-45 right triangle is a units long, then the

hypotenuse is a √ 2 units long.45°45°

aa a √ 2a √ 2

45°45°aa

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

Find the length of the hypotenuse in the 45-45 right triangle. 45°45°

33 xx

45°45°33

3 √ 23 √ 2

Example 1Example 1

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

45°45°

aa

45°45°aa

5 √ 25 √ 2

55

Example 2Example 2

If a = 2, what is c?If a = 2, what is c?45°45°

aa cc

45°45°bb

2 √ 22 √ 2

ExampleExample

If c = 8, what is a?If c = 8, what is a?45°45°

aa cc

45°45°bb

4 √ 24 √ 2

ExampleExample

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

45°45°

aa cc

45°45°bb

a = 3 √ 2 and c = 6a = 3 √ 2 and c = 6

ExampleExample

If the perimeter is 20, what is a?If the perimeter is 20, what is a?

45°45°

aa cc

45°45°bb

≈ 5.86≈ 5.86

ExampleExample

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

nono

ExampleExample

60°60°

60°60° 60°60°

30°30°30°30°

30-60 Right Triangle30-60 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 2a units long.

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 2a units long.

60°60°2a2a

a √ 3a √ 3 30°30°

aa

Find the missing lengths x and y in the triangle.Find the missing lengths x and y in the triangle.

60°60°

44xx

30°30° yy

Since a = 4, the long leg is 4 √ 3 and the hypotenuse is

2(4) = 8 units.

Since a = 4, the long leg is 4 √ 3 and the hypotenuse is

2(4) = 8 units.

Example 3Example 3

Find the short leg and hypotenuse of a 30-60 right Δ whose long leg is 6 √ 3.

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 2a = 2(6) = 12.

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 2a = 2(6) = 12.

Example 4Example 4

Find the long leg and short leg of a 30-60 right triangle whose hypotenuse is 9.

Find the long leg and short leg of a 30-60 right triangle whose hypotenuse is 9.

The hypotenuse is 2a = 9, so the short leg is a = 4.5. The

long leg is a √ 3 = 4.5 √ 3.

The hypotenuse is 2a = 9, so the short leg is a = 4.5. The

long leg is a √ 3 = 4.5 √ 3.

Example 5Example 5

If a = 1, what are b and c?If a = 1, what are b and c?

60°60°

aacc

30°30°bb

b = √ 3 and c = 2b = √ 3 and c = 2

ExampleExample

If c = 7, what is a?If c = 7, what is a?

60°60°

aacc

30°30°bb

7272

= 3.5= 3.5

ExampleExample

If a = 3, what is the perimeter?If a = 3, what is the perimeter?

60°60°

aacc

30°30°bb

9 + 3 √ 3 ≈ 14.29 + 3 √ 3 ≈ 14.2

ExampleExample

If the perimeter is 25, what is c?If the perimeter is 25, what is c? 60°60°

aacc

30°30°bb

10.56610.566

ExampleExample

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?

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)no; 5 ≠ 2(3)

ExampleExample

Find the other two sides in a 30-60 right triangle with a short leg of √ 5.

Find the other two sides in a 30-60 right triangle with a short leg of √ 5.

long leg: √ 15; hypotenuse: 2 √ 5sides: 2.2, 3.9, 4.5

long leg: √ 15; hypotenuse: 2 √ 5sides: 2.2, 3.9, 4.5

ExerciseExercise

Find the other two sides in a 30-60 right triangle with a short leg of 2 √ 7.

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

long leg: 2 √ 21; hypotenuse: 4 √ 7

sides: 5.3, 9.2, 10.6

ExerciseExercise

Find the other two sides in a 30-60 right triangle with a long leg of 20 √ 3.

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

short leg: 20; hypotenuse: 40

sides: 20, 34.6, 40

ExerciseExercise