The lengths of two sides of a triangle are 8 cm and 9 cm, and its area is…
2023
The lengths of two sides of a triangle are 8 cm and 9 cm, and its area is 12√5cm². What is the length of the third side?
- A.
5 cm
- B.
9 cm
- C.
7 cm
- D.
More than one of the above
- E.
None of the above
Attempted by 2 students.
Show answer & explanation
Correct answer: C
Step 1: Use the area formula for a triangle: Area = 1/2 * a * b * sin(C). Given Area = 12√5, a = 8, b = 9.
Step 2: Solve for sin(C): 12√5 = 1/2 * 8 * 9 * sin(C) => 12√5 = 36 * sin(C) => sin(C) = √5 / 3.
Step 3: Find cos(C) using sin²(C) + cos²(C) = 1. cos²(C) = 1 - (5/9) = 4/9 => cos(C) = ±2/3.
Step 4: Apply the Law of Cosines: c² = a² + b² - 2ab * cos(C). Case 1 (cos(C) = 2/3): c² = 64 + 81 - 96 = 49 => c = 7.
Step 5: Case 2 (cos(C) = -2/3): c² = 64 + 81 + 96 = 241 => c = √241. Only 7 cm matches the given options.