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Answer :
To find the area of the triangle with sides [tex]\(a = 25\)[/tex], [tex]\(b = 13\)[/tex], and [tex]\(c = 17\)[/tex], we can use Heron's formula.
Heron's formula states that the area of a triangle with side lengths [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] is given by:
[tex]\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \][/tex]
where [tex]\(s\)[/tex] is the semi-perimeter of the triangle:
[tex]\[ s = \frac{a + b + c}{2} \][/tex]
Let's calculate it step by step:
1. Find the semi-perimeter [tex]\(s\)[/tex]:
[tex]\[
s = \frac{25 + 13 + 17}{2} = \frac{55}{2} = 27.5
\][/tex]
2. Apply Heron's formula to find the area:
[tex]\[
\text{Area} = \sqrt{27.5 \times (27.5 - 25) \times (27.5 - 13) \times (27.5 - 17)}
\][/tex]
3. Calculate the individual terms:
[tex]\[
27.5 - 25 = 2.5
\][/tex]
[tex]\[
27.5 - 13 = 14.5
\][/tex]
[tex]\[
27.5 - 17 = 10.5
\][/tex]
4. Substitute these values back into the formula:
[tex]\[
\text{Area} = \sqrt{27.5 \times 2.5 \times 14.5 \times 10.5}
\][/tex]
5. Calculate the expression under the square root:
[tex]\[
\text{Area} = \sqrt{27.5 \times 2.5 \times 14.5 \times 10.5} = \sqrt{96206.25}
\][/tex]
6. Find the square root:
[tex]\[
\text{Area} \approx 99.1
\][/tex]
After evaluating the steps, we find the area is approximately 99.1 square units.
Thus, the best choice from the given options is:
- A: 99.1 units²
Heron's formula states that the area of a triangle with side lengths [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] is given by:
[tex]\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \][/tex]
where [tex]\(s\)[/tex] is the semi-perimeter of the triangle:
[tex]\[ s = \frac{a + b + c}{2} \][/tex]
Let's calculate it step by step:
1. Find the semi-perimeter [tex]\(s\)[/tex]:
[tex]\[
s = \frac{25 + 13 + 17}{2} = \frac{55}{2} = 27.5
\][/tex]
2. Apply Heron's formula to find the area:
[tex]\[
\text{Area} = \sqrt{27.5 \times (27.5 - 25) \times (27.5 - 13) \times (27.5 - 17)}
\][/tex]
3. Calculate the individual terms:
[tex]\[
27.5 - 25 = 2.5
\][/tex]
[tex]\[
27.5 - 13 = 14.5
\][/tex]
[tex]\[
27.5 - 17 = 10.5
\][/tex]
4. Substitute these values back into the formula:
[tex]\[
\text{Area} = \sqrt{27.5 \times 2.5 \times 14.5 \times 10.5}
\][/tex]
5. Calculate the expression under the square root:
[tex]\[
\text{Area} = \sqrt{27.5 \times 2.5 \times 14.5 \times 10.5} = \sqrt{96206.25}
\][/tex]
6. Find the square root:
[tex]\[
\text{Area} \approx 99.1
\][/tex]
After evaluating the steps, we find the area is approximately 99.1 square units.
Thus, the best choice from the given options is:
- A: 99.1 units²
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