Question

find the area of this triangle.
32
48
63
? ft²
round to the nearest whole number.

Explanation:

Step1: Use Heron's formula. First, find the semi - perimeter \(s\).

The sides of the triangle are \(a = 32\), \(b = 48\), \(c = 63\). The formula for semi - perimeter is \(s=\frac{a + b + c}{2}\).
\(s=\frac{32+48 + 63}{2}=\frac{143}{2}=71.5\)

Step2: Apply Heron's formula for area \(A=\sqrt{s(s - a)(s - b)(s - c)}\).

Substitute \(s = 71.5\), \(a = 32\), \(b = 48\), \(c = 63\) into the formula:
\(A=\sqrt{71.5\times(71.5 - 32)\times(71.5 - 48)\times(71.5 - 63)}\)
\(=\sqrt{71.5\times39.5\times23.5\times8.5}\)
First, calculate the product inside the square root:
\(71.5\times39.5 = 71.5\times(40 - 0.5)=71.5\times40-71.5\times0.5 = 2860 - 35.75 = 2824.25\)
\(23.5\times8.5=(20 + 3.5)\times8.5=20\times8.5+3.5\times8.5 = 170+29.75 = 199.75\)
Then, \(2824.25\times199.75=(2824 + 0.25)\times(200 - 0.25)=2824\times200-2824\times0.25+0.25\times200 - 0.25\times0.25\)
\(=564800-706 + 50-0.0625=564800+50-(706 + 0.0625)=564850 - 706.0625 = 564143.9375\)
Now, \(A=\sqrt{564143.9375}\approx751\)

Answer:

\(751\)