To solve for the area of a triangle with two given sides, [tex]\( a = 12 \)[/tex] inches and [tex]\( b = 14 \)[/tex] inches, and a given perimeter of 34 inches, follow these steps:
1. Determine the length of the third side:
Given the perimeter [tex]\( P \)[/tex] of the triangle is 34 inches, we have:
[tex]\[
P = a + b + c
\][/tex]
Substituting the known values:
[tex]\[
34 = 12 + 14 + c
\][/tex]
Solving for [tex]\( c \)[/tex]:
[tex]\[
c = 34 - 12 - 14 = 8 \text{ inches}
\][/tex]
2. Calculate the semi-perimeter [tex]\( s \)[/tex]:
The semi-perimeter of a triangle is half of the perimeter:
[tex]\[
s = \frac{P}{2}
\][/tex]
Substituting the known perimeter:
[tex]\[
s = \frac{34}{2} = 17 \text{ inches}
\][/tex]
3. Use Heron's formula to find the area:
Heron's formula for the area [tex]\( A \)[/tex] of a triangle is given by:
[tex]\[
A = \sqrt{s(s - a)(s - b)(s - c)}
\][/tex]
Substituting the values of [tex]\( s \)[/tex], [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
A = \sqrt{17 \times (17 - 12) \times (17 - 14) \times (17 - 8)}
\][/tex]
[tex]\[
A = \sqrt{17 \times 5 \times 3 \times 9}
\][/tex]
Calculating the values inside the square root:
[tex]\[
A = \sqrt{17 \times 5 \times 3 \times 9} = \sqrt{2295}
\][/tex]
4. Determine the numerical value of the area:
Calculating the square root of 2295:
[tex]\[
A \approx 47.91 \text{ square inches}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{47.91 \text{ in}^2}
\][/tex]
This matches option D in the provided choices.
