Question

find the area of a triangular garden if the sides are approximately 4 feet, 12 feet, and 13 feet. round to the nearest square foot.

a. 442ft²
b. 24ft²
c. 6ft²
d. 312ft²

Explanation:

Step1: Calculate the semi - perimeter \(s\)

The formula for the semi - perimeter of a triangle with side lengths \(a\), \(b\), and \(c\) is \(s=\frac{a + b + c}{2}\).
Given \(a = 4\), \(b = 12\), \(c = 13\), then \(s=\frac{4+12 + 13}{2}=\frac{29}{2}=14.5\).

Step2: Apply Heron's formula

Heron's formula for the area \(A\) of a triangle is \(A=\sqrt{s(s - a)(s - b)(s - c)}\).
Substitute \(s = 14.5\), \(a = 4\), \(b = 12\), \(c = 13\) into the formula:
\(A=\sqrt{14.5(14.5 - 4)(14.5 - 12)(14.5 - 13)}\)
\(=\sqrt{14.5\times10.5\times2.5\times1.5}\)
\(=\sqrt{14.5\times(10.5\times2.5\times1.5)}\)
\(=\sqrt{14.5\times39.375}\)
\(=\sqrt{570.9375}\approx24\)

Answer:

B. \(24\mathrm{ft}^2\)