Question

find the area of the triangle.
a 21
b 15
c 32

Explanation:

Step1: Calculate semi-perimeter

First, find the semi-perimeter \(s\) of the triangle using the formula for semi-perimeter of a triangle with sides \(a, b, c\):

$$ s = \frac{a + b + c}{2} = \frac{21 + 15 + 32}{2} = \frac{68}{2} = 34 $$

Step2: Apply Heron's Formula

Use Heron's formula to find the area \(A\), where \(A = \sqrt{s(s-a)(s-b)(s-c)}\):

$$ A = \sqrt{34(34-21)(34-15)(34-32)} = \sqrt{34 \times 13 \times 19 \times 2} $$

Step3: Compute the product inside root

Calculate the product step by step:

$$ 34 \times 2 = 68, \quad 13 \times 19 = 247, \quad 68 \times 247 = 16796 $$

Step4: Calculate square root

Find the square root of the product:

$$ A = \sqrt{16796} \approx 129.6 $$

Answer:

The area of the triangle is approximately 129.6 square units (or exactly $\sqrt{16796}$ square units).