Question

find the area of triangle abc with the given parts. round to the nearest square centimeter.
a = 19.5 cm
b = 14.5 cm
c = 14.9 cm

a. 113 cm²
b. 116 cm²
c. 110 cm²
d. 107 cm²

Explanation:

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

The formula for the semi - perimeter of a triangle is \( s=\frac{a + b + c}{2} \).
Given \( a = 19.5\mathrm{cm}\), \( b = 14.5\mathrm{cm}\), \( c = 14.9\mathrm{cm}\), then \( s=\frac{19.5+14.5 + 14.9}{2}=\frac{48.9}{2}=24.45\mathrm{cm}\)

Step2: Use Heron's formula to find the area \( A \)

Heron's formula is \( A=\sqrt{s(s - a)(s - b)(s - c)} \)
Substitute the values:
\( s - a=24.45-19.5 = 4.95\), \( s - b=24.45-14.5 = 9.95\), \( s - c=24.45-14.9 = 9.55\)
\( A=\sqrt{24.45\times4.95\times9.95\times9.55}\)
\( A=\sqrt{24.45\times(4.95\times9.95\times9.55)}\)
First calculate \(4.95\times9.95\times9.55=(5 - 0.05)\times9.95\times9.55=(49.75-0.4975)\times9.55 = 49.2525\times9.55=49.2525\times(9 + 0.5+0.05)=443.2725+24.62625 + 2.462625 = 470.361375\)
Then \(24.45\times470.361375=24\times470.361375+0.45\times470.361375=11288.673+211.66261875 = 11500.33561875\)
\( A=\sqrt{11500.33561875}\approx107.24\approx107\mathrm{cm}^2\)

Answer:

D. \(107\mathrm{cm}^2\)