Answer :
To find the area of a triangle with sides [tex]\( a = 25 \)[/tex], [tex]\( b = 13 \)[/tex], and [tex]\( c = 17 \)[/tex], we can use Heron's formula. Here's a step-by-step approach:
1. Calculate the Semi-perimeter (s):
The semi-perimeter [tex]\( s \)[/tex] is half of the sum of the sides of the triangle.
[tex]\[
s = \frac{a + b + c}{2} = \frac{25 + 13 + 17}{2} = \frac{55}{2} = 27.5
\][/tex]
2. Apply Heron's Formula:
Heron's formula states that the area [tex]\( A \)[/tex] of a triangle with semi-perimeter [tex]\( s \)[/tex] and sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] is:
[tex]\[
A = \sqrt{s(s - a)(s - b)(s - c)}
\][/tex]
Substituting the values into the formula:
[tex]\[
A = \sqrt{27.5 \times (27.5 - 25) \times (27.5 - 13) \times (27.5 - 17)}
\][/tex]
[tex]\[
A = \sqrt{27.5 \times 2.5 \times 14.5 \times 10.5}
\][/tex]
Calculate the area:
[tex]\[
A \approx \sqrt{10251.5} \approx 102.3
\][/tex]
3. Select the Closest Answer:
Comparing the calculated area with the options provided:
- a. 99.1 units²
- b. 100.5 units²
- c. 98.7 units²
- d. 102.3 units²
The calculated area, 102.3, is closest to option D.
So, the best choice is:
D. 102.3 units²
1. Calculate the Semi-perimeter (s):
The semi-perimeter [tex]\( s \)[/tex] is half of the sum of the sides of the triangle.
[tex]\[
s = \frac{a + b + c}{2} = \frac{25 + 13 + 17}{2} = \frac{55}{2} = 27.5
\][/tex]
2. Apply Heron's Formula:
Heron's formula states that the area [tex]\( A \)[/tex] of a triangle with semi-perimeter [tex]\( s \)[/tex] and sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] is:
[tex]\[
A = \sqrt{s(s - a)(s - b)(s - c)}
\][/tex]
Substituting the values into the formula:
[tex]\[
A = \sqrt{27.5 \times (27.5 - 25) \times (27.5 - 13) \times (27.5 - 17)}
\][/tex]
[tex]\[
A = \sqrt{27.5 \times 2.5 \times 14.5 \times 10.5}
\][/tex]
Calculate the area:
[tex]\[
A \approx \sqrt{10251.5} \approx 102.3
\][/tex]
3. Select the Closest Answer:
Comparing the calculated area with the options provided:
- a. 99.1 units²
- b. 100.5 units²
- c. 98.7 units²
- d. 102.3 units²
The calculated area, 102.3, is closest to option D.
So, the best choice is:
D. 102.3 units²
