Question

what is the area of triangle abc? round to the nearest hundredth of a square unit.
17.75 square units
81.33 square units
372.71 square units
957.74 square units
herons formula: area = \sqrt{s(s - a)(s - b)(s - c)}

Explanation:

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

The sides of the triangle are \( a = 14 \), \( b = 16 \), \( c = 12 \). The formula for the semi - perimeter \( s=\frac{a + b + c}{2}\).
Substitute the values: \( s=\frac{14 + 16+12}{2}=\frac{42}{2} = 21 \)

Step2: Apply Heron's formula

Heron's formula is \( \text{Area}=\sqrt{s(s - a)(s - b)(s - c)} \)
Substitute \( s = 21 \), \( a = 14 \), \( b = 16 \), \( c = 12 \) into the formula:
\( \text{Area}=\sqrt{21(21 - 14)(21 - 16)(21 - 12)}=\sqrt{21\times7\times5\times9} \)
First, calculate the product inside the square root: \( 21\times7 = 147 \), \( 5\times9=45 \), then \( 147\times45 = 6615 \)
Then, \( \text{Area}=\sqrt{6615}\approx81.33 \)

Answer:

81.33 square units