Question

The sides of a triangle are 13, 14, 15. Find the area.

Hint:

Heron's formula is the go-to method for finding the area of a triangle when only the three side lengths are provided.

Answer 1

Correct Answer: 84

Step 1: Understanding the Question:
We are given the lengths of the three sides of a triangle and asked to find its area. Since the sides are of different lengths, this is a scalene triangle.
Step 2: Key Formula or Approach:
We can use Heron's formula to find the area of a triangle when all three side lengths are known. The formula is: \[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \] where a, b, and c are the side lengths, and s is the semi-perimeter of the triangle, calculated as: \[ s = \frac{a+b+c}{2} \] Step 3: Detailed Explanation:
First, calculate the semi-perimeter (s):
Given sides a = 13, b = 14, c = 15. \[ s = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21 \] Now, apply Heron's formula: \[ \text{Area} = \sqrt{21(21-13)(21-14)(21-15)} \] \[ \text{Area} = \sqrt{21 \times 8 \times 7 \times 6} \] To simplify the calculation, break down the numbers into their prime factors: \[ \text{Area} = \sqrt{(3 \times 7) \times (2^3) \times 7 \times (2 \times 3)} \] \[ \text{Area} = \sqrt{3^2 \times 7^2 \times 2^4} \] Now, take the square root of each term: \[ \text{Area} = 3 \times 7 \times 2^2 = 21 \times 4 = 84 \] Step 4: Final Answer:
The area of the triangle is 84 square units.