Solutions Triangle exercises & problems

The solutions to the 12 triangle exercises and problems:

 

Basic Level:

1. Calculate the Perimeter:
   P = 9 inches + 12 inches + 15 inches = 36 inches.

2. Length of Hypotenuse in a Right Triangle:
   Using the Pythagorean Theorem:
   c² = 6 cm² + 8 cm² = 36 cm² + 64 cm² = 100 cm².
   c = √100 cm² = 10 cm.

3. Area of an Equilateral Triangle:
   A = (side length² * √3) / 4 = (10 m)² * √3 / 4 ≈ 43.3 square meters.

4. Area of an Isosceles Triangle:
   A = (base * height) / 2 = (6 inches * 8 inches) / 2 = 24 square inches.

 

Intermediate Level:

5. Find Side Lengths Given Angles and Side:
   Using the Law of Sines:

   • For side a: a / sin(A) = c / sin(C), so a / sin(40°) = 12 ft / sin(65°), giving a ≈ 9.47 ft.
   • For side b: b / sin(B) = c / sin(C), so b / sin(65°) = 12 ft / sin(40°), giving b ≈ 14.57 ft.

6. Area of Right Triangle:
   A = (1/2) * base * height = (1/2) * 15 yards * 20 yards = 150 square yards.

7. Length of Median:
   Using the formula for the length of a median from vertex B: Length of Median = (1/2) * √(2a² + 2c² - b²) Length of Median ≈ (1/2) * √(2(7 in)² + 2(10 in)² - (9 in)²) ≈ 5.77 inches.

8. Circumradius (R):
   Using the formula for the circumradius: R = (abc) / (4A) = (5 ft * 12 ft * 13 ft) / (4 * 30 square feet) = 13 ft / 4 ≈ 3.25 feet.

 

Advanced Level:

9. Length of Side c Given Angle C:
   Using the Law of Sines: c / sin(C) = a / sin(A), so c / sin(30°) = 8 cm / sin(40°), giving c ≈ 9.77 cm.

10. Find Side Lengths Given Angles and Side:
    Using the Law of Sines:

    • For side b: b / sin(B) = a / sin(A), so b / sin(75°) = 10 m / sin(60°), giving b ≈ 12.32 m.
    • For side c: c / sin(C) = a / sin(A), so c / sin(60°) = 10 m / sin(75°), giving c ≈ 20 m.

11. Inradius (r):
    Using the formula for inradius: r = A / s, where s is the semiperimeter. Calculate the semiperimeter first:
    s = (a + b + c) / 2 = (7 in + 24 in + 25 in) / 2 = 28 in / 2 = 14 in. Then, calculate the inradius: r = 84 in² / 14 in = 6 inches.

12. Measure of Angle A Using the Law of Cosines:
    Using the Law of Cosines: cos(A) = (b² + c² - a²) / (2bc)
    cos(A) = (8 inches)² + (15 inches)² - (17 inches)²) / (2 * 8 inches * 15 inches)
    cos(A) = (64 + 225 - 289) / (2 * 120)
    cos(A) = 0.33
    A ≈ arccos(0.33) ≈ 71.8 degrees.