Question

the sides of a triangle measure 32 m, 36 m, and 37 m. find its area. write your answer as an integer or as a decimal rounded to the nearest tenth. m² submit

Explanation:

Step1: Calculate the semi - perimeter (s)

The formula for the semi - perimeter of a triangle with sides \(a\), \(b\), \(c\) is \(s=\frac{a + b + c}{2}\). Given \(a = 32\), \(b = 36\), \(c = 37\), we have:
\(s=\frac{32+36 + 37}{2}=\frac{105}{2}=52.5\)

Step2: Apply Heron's formula

Heron's formula for the area of a triangle is \(A=\sqrt{s(s - a)(s - b)(s - c)}\).
Substitute \(s = 52.5\), \(a = 32\), \(b = 36\), \(c = 37\) into the formula:
\(A=\sqrt{52.5\times(52.5 - 32)\times(52.5 - 36)\times(52.5 - 37)}\)
First, calculate the values inside the square root:
\(52.5-32 = 20.5\)
\(52.5 - 36=16.5\)
\(52.5 - 37 = 15.5\)
Then, \(A=\sqrt{52.5\times20.5\times16.5\times15.5}\)
Calculate \(52.5\times20.5 = 52.5\times(20 + 0.5)=52.5\times20+52.5\times0.5 = 1050+26.25 = 1076.25\)
\(16.5\times15.5=(16 + 0.5)\times(15+0.5)=16\times15+16\times0.5+0.5\times15 + 0.5\times0.5=240 + 8+7.5+0.25 = 255.75\)
Now, multiply these two results: \(1076.25\times255.75 = 1076.25\times(250+5.75)=1076.25\times250+1076.25\times5.75\)
\(1076.25\times250 = 269062.5\)
\(1076.25\times5.75=1076.25\times(5 + 0.75)=1076.25\times5+1076.25\times0.75 = 5381.25+807.1875 = 6188.4375\)
\(269062.5+6188.4375 = 275250.9375\)
Then, \(A=\sqrt{275250.9375}\approx524.6\)

Answer:

\(524.6\)