Special Triangles In Real Life

1) A ladder leans against a wall forming an angle of π/6 with the ground. If the ladder is 12 m long, how high up the wall does it reach?

6 m

4√3 m

12√3 m

2√3 m

Height = 12·sin(π/6) = 12·(1/2) = 6 m.

Explanation

Height = 12·sin(π/6) = 12·(1/2) = 6 m.

2) A ramp makes an angle of π/4 with the ground. If the ramp length is 8 m, what is the vertical rise?

4√2 m

8 m

2√2 m

6√2 m

Rise = 8·sin(π/4) = 8·(√2/2) = 4√2 m.

Explanation

Rise = 8·sin(π/4) = 8·(√2/2) = 4√2 m.

3) A beam is placed at an angle of π/3 with the ground. If the beam is 10 m long, how far is its base from the wall?

10/√3 m

10√3 m

5 m

15 m

Base = 10·cos(π/3) = 10·(1/2) = 5 m.

Explanation

Base = 10·cos(π/3) = 10·(1/2) = 5 m.

4) A ladder is 15 m long and leans to form an angle of π/4 with the wall. How far is the base from the wall?

15/√2 m

10 m

12√2 m

20 m

Angle with wall is 45°, so base = 15·cos(45°) = 15/√2 m.

Explanation

Angle with wall is 45°, so base = 15·cos(45°) = 15/√2 m.

5) A triangular support forms a 30° angle with the ground. If the vertical side is 5 m, what is the hypotenuse length?

7.5 m

5√3 m

5/√3 m

10 m

Hypotenuse = opposite/sin30° = 5/(1/2) = 10 m.

Explanation

Hypotenuse = opposite/sin30° = 5/(1/2) = 10 m.

6) A board is placed making a 60° angle with the floor. If the vertical height reached is 8 m, how long is the board?

12 m

16/√3 m

8/√3 m

8√3 m

Length = 8/sin60° = 8/(√3/2) = 16/√3 m.

Explanation

Length = 8/sin60° = 8/(√3/2) = 16/√3 m.

7) A square garden diagonal measures 6√2 m. What is the side length of the garden?

3 m

6 m

9 m

12 m

For a square, d = s√2 ⇒ s = d/√2 = (6√2)/√2 = 6 m.

Explanation

For a square, d = s√2 ⇒ s = d/√2 = (6√2)/√2 = 6 m.

8) A ladder forms a 45° angle with the ground and touches the wall at 7 m height. What is the ladder length?

7√2 m

14 m

10 m

5√3 m

Length = 7/sin45° = 7·√2 m.

Explanation

Length = 7/sin45° = 7·√2 m.

9) A triangular ramp has a hypotenuse of 20 m and forms a 60° angle with the base. What is the base length?

15 m

20/√3 m

20√3/2 m

10 m

Base = 20·cos60° = 20·(1/2) = 10 m.

Explanation

Base = 20·cos60° = 20·(1/2) = 10 m.

10) A support forms a 30° angle with the ground. If the base length is 9 m, what is the vertical rise?

3√3 m

6 m

9/√3 m

12√3 m

Rise = 9·tan30° = 9/√3 = 3√3 m.

Explanation

Rise = 9·tan30° = 9/√3 = 3√3 m.

11) A construction beam is 18 m long and forms a 45° angle with the ground. What is its vertical height?

12√2 m

9 m

18/√2 m

15 m

Use sin(45°) = 1/√2; height = 18 × (1/√2) = 18/√2 ≈ 12.7 m.

Explanation

Use sin(45°) = 1/√2; height = 18 × (1/√2) = 18/√2 ≈ 12.7 m.

12) A triangular sail has a right angle, with one angle measuring π/3. If the shortest side is 5 m, what is the hypotenuse?

10 m

5√3 m

15 m

7.5 m

In a 30°–60°–90° triangle, the hypotenuse = 2 × shortest side = 10 m.

Explanation

In a 30°–60°–90° triangle, the hypotenuse = 2 × shortest side = 10 m.

13) A square tile has a diagonal of 10√2 cm. What is the side length?

20 cm

5√2 cm

10 cm

15 cm

Diagonal = side × √2, so side = (10√2)/√2 = 10 cm.

Explanation

Diagonal = side × √2, so side = (10√2)/√2 = 10 cm.

14) A ramp makes an angle of 60° with the ground. If the ramp is 12 m long, what is the vertical rise?

6√3 m

12√3 m

6 m

9

Vertical height = 12 × sin(60°) = 12 × √3/2 = 6√3 m.

Explanation

Vertical height = 12 × sin(60°) = 12 × √3/2 = 6√3 m.

15) A board leans against a wall forming a 45° angle. If the base is 4 m from the wall, how high does the board reach?

4√2 m

8 m

4 m

2√2 m

For 45°, height = base = 4 m (since opposite = adjacent in 45° triangles).

Explanation

For 45°, height = base = 4 m (since opposite = adjacent in 45° triangles).

16) A triangular flag has sides in the ratio of a 30°–60°–90° triangle. If the shortest side is 2 m, what is the longest side?

3 m

2√3 m

6 m

4 m

In a 30°–60°–90° triangle, the longest side = 2 × shortest side = 4 m.

Explanation

In a 30°–60°–90° triangle, the longest side = 2 × shortest side = 4 m.

17) A rope forms a 30° angle with the ground and is 24 m long. What is the vertical height reached?

12 m

24√3 m

12√3 m

8 m

Height = 24 × sin(30°) = 24 × 0.5 = 12 m.

Explanation

Height = 24 × sin(30°) = 24 × 0.5 = 12 m.

18) A rectangular park measures 20 m by 20 m. What is the length of its diagonal?

10√2 m

30 m

40 m

20√2 m

Diagonal = side × √2 = 20√2 m.

Explanation

Diagonal = side × √2 = 20√2 m.

19) A scaffold is placed at 60° to the ground. If the horizontal distance from the wall is 5 m, what is the scaffold’s length?

10 m

5√3 m

5/√3 m

15 m

cos(60°) = base/hypotenuse ⇒ hypotenuse = base / cos(60°) = 5 / 0.5 = 10 m.

Explanation

cos(60°) = base/hypotenuse ⇒ hypotenuse = base / cos(60°) = 5 / 0.5 = 10 m.

20) A square playground has a side length of 14 m. What is the length of its diagonal?

14√2 m

28 m

7 m

21√2 m

Diagonal = side × √2 = 14√2 m.

Explanation

Diagonal = side × √2 = 14√2 m.

Special Triangles in Real Life