Solved AAI Worksheet #2 - Solving Right Solve each triangle ... - Free Printable

Let’s solve each triangle one by one. We’ll use the Pythagorean Theorem (a² + b² = c²) for right triangles when we have two sides, and trig ratios (sin, cos, tan) when we have an angle and a side.

We’re told to round answers to the nearest tenth.

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Problem 1:
Right triangle ABC, right angle at C.
AC = 9, BC = 12 → find AB (hypotenuse), and angles A and B.

Step 1: Find hypotenuse AB using Pythagoras:
AB² = AC² + BC² = 9² + 12² = 81 + 144 = 225
→ AB = √225 = 15.0

Step 2: Find angle B (at vertex B):
tan(B) = opposite/adjacent = AC / BC = 9 / 12 = 0.75
→ B = arctan(0.75) ≈ 36.9°

Step 3: Angle A = 90° - 36.9° = 53.1°

✔ Answer: AB = 15.0, ∠A = 53.1°, ∠B = 36.9°

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Problem 2:
Triangle ABC, right angle at C.
Angle A = 38°, side BC = 12.5 → find AB (hypotenuse), AC, and angle B.

Step 1: Angle B = 90° - 38° = 52°

Step 2: Use sin(A) = opposite/hypotenuse = BC / AB
sin(38°) = 12.5 / AB
→ AB = 12.5 / sin(38°) ≈ 12.5 / 0.6157 ≈ 20.3

Step 3: Use tan(A) = opposite/adjacent = BC / AC
tan(38°) = 12.5 / AC
→ AC = 12.5 / tan(38°) ≈ 12.5 / 0.7813 ≈ 16.0

✔ Answer: AB = 20.3, AC = 16.0, ∠B = 52.0°

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Problem 3:
Triangle ABC, right angle at C.
AC = 2, AB = 3 → find BC (side a), and angles A and B.

Step 1: Use Pythagoras: AB² = AC² + BC²
3² = 2² + a² → 9 = 4 + a² → a² = 5 → a = √5 ≈ 2.2

Step 2: Find angle A:
cos(A) = adjacent/hypotenuse = AC / AB = 2 / 3 ≈ 0.6667
→ A = arccos(0.6667) ≈ 48.2°

Step 3: Angle B = 90° - 48.2° = 41.8°

✔ Answer: BC = 2.2, ∠A = 48.2°, ∠B = 41.8°

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Problem 4:
Triangle ABC, right angle at C.
Angle A = 59°, hypotenuse AB = 16 → find AC, BC, and angle B.

Step 1: Angle B = 90° - 59° = 31°

Step 2: AC = adjacent to angle A → cos(59°) = AC / 16
→ AC = 16 * cos(59°) ≈ 16 * 0.5150 ≈ 8.2

Step 3: BC = opposite to angle A → sin(59°) = BC / 16
→ BC = 16 * sin(59°) ≈ 16 * 0.8572 ≈ 13.7

✔ Answer: AC = 8.2, BC = 13.7, ∠B = 31.0°

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Problem 5:
Triangle ABC, right angle at C.
AC = 6, BC = 12 → find AB (c), and angles A and B.

Step 1: AB² = 6² + 12² = 36 + 144 = 180 → AB = √180 ≈ 13.4

Step 2: tan(A) = opposite/adjacent = BC / AC = 12 / 6 = 2
→ A = arctan(2) ≈ 63.4°

Step 3: B = 90° - 63.4° = 26.6°

✔ Answer: AB = 13.4, ∠A = 63.4°, ∠B = 26.6°

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Problem 6:
Triangle ABC, right angle at C.
Angle A = 61°, side AC = 5 → find AB, BC, and angle B.

Step 1: Angle B = 90° - 61° = 29°

Step 2: AB = hypotenuse → cos(61°) = AC / AB = 5 / AB
→ AB = 5 / cos(61°) ≈ 5 / 0.4848 ≈ 10.3

Step 3: BC = opposite → tan(61°) = BC / 5
→ BC = 5 * tan(61°) ≈ 5 * 1.8040 ≈ 9.0

✔ Answer: AB = 10.3, BC = 9.0, ∠B = 29.0°

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Problem 7:
Triangle ABC, right angle at C.
Angle B = 51°, side BC = 3 → find AB (c), AC (b), and angle A.

Step 1: Angle A = 90° - 51° = 39°

Step 2: AB = hypotenuse → cos(51°) = BC / AB = 3 / AB
→ AB = 3 / cos(51°) ≈ 3 / 0.6293 ≈ 4.8

Step 3: AC = opposite → tan(51°) = AC / 3
→ AC = 3 * tan(51°) ≈ 3 * 1.2349 ≈ 3.7

✔ Answer: AB = 4.8, AC = 3.7, ∠A = 39.0°

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Problem 8:
Triangle ABC, right angle at C.
Angle B = 52°, side AC = 2 → find AB, BC, and angle A.

Step 1: Angle A = 90° - 52° = 38°

Step 2: AB = hypotenuse → sin(52°) = AC / AB = 2 / AB
→ AB = 2 / sin(52°) ≈ 2 / 0.7880 ≈ 2.5

Step 3: BC = adjacent → tan(52°) = AC / BC = 2 / BC
→ BC = 2 / tan(52°) ≈ 2 / 1.2799 ≈ 1.6

✔ Answer: AB = 2.5, BC = 1.6, ∠A = 38.0°

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Problem 9:
Triangle ABC, right angle at C.
AC = 3, BC = 3 → isosceles right triangle.

Step 1: AB² = 3² + 3² = 9 + 9 = 18 → AB = √18 ≈ 4.2

Step 2: Since legs are equal, angles A and B are both 45°.

✔ Answer: AB = 4.2, ∠A = 45.0°, ∠B = 45.0°

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Problem 10:
Triangle ABC, right angle at C.
Angle A = 65°, hypotenuse AB = 13 → find AC, BC, and angle B.

Step 1: Angle B = 90° - 65° = 25°

Step 2: AC = adjacent → cos(65°) = AC / 13
→ AC = 13 * cos(65°) ≈ 13 * 0.4226 ≈ 5.5

Step 3: BC = opposite → sin(65°) = BC / 13
→ BC = 13 * sin(65°) ≈ 13 * 0.9063 ≈ 11.8

✔ Answer: AC = 5.5, BC = 11.8, ∠B = 25.0°

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Final Answer:

1) AB = 15.0, ∠A = 53.1°, ∠B = 36.9°
2) AB = 20.3, AC = 16.0, ∠B = 52.0°
3) BC = 2.2, ∠A = 48.2°, ∠B = 41.8°
4) AC = 8.2, BC = 13.7, ∠B = 31.0°
5) AB = 13.4, ∠A = 63.4°, ∠B = 26.6°
6) AB = 10.3, BC = 9.0, ∠B = 29.0°
7) AB = 4.8, AC = 3.7, ∠A = 39.0°
8) AB = 2.5, BC = 1.6, ∠A = 38.0°
9) AB = 4.2, ∠A = 45.0°, ∠B = 45.0°
10) AC = 5.5, BC = 11.8, ∠B = 25.0°