Question
Find the area of the triangle
B
14 in
C
9 in
A
13 in
B
14 in
C
9 in
A
13 in
Answers
Answered by
GPT-4o mini
To find the area of a triangle when the lengths of all three sides are known, we can use Heron's formula. Given the lengths of the sides \(a = 14\) in, \(b = 9\) in, and \(c = 13\) in, we proceed with the following steps:
1. **Calculate the semi-perimeter (s):**
\[
s = \frac{a + b + c}{2} = \frac{14 + 9 + 13}{2} = \frac{36}{2} = 18 \text{ in}
\]
2. **Apply Heron's formula:**
The area \(A\) of the triangle can be calculated using the formula:
\[
A = \sqrt{s(s - a)(s - b)(s - c)}
\]
Substitute \(s\) and the side lengths into the formula:
\[
A = \sqrt{18(18 - 14)(18 - 9)(18 - 13)}
\]
\[
A = \sqrt{18 \times 4 \times 9 \times 5}
\]
3. **Calculate the products:**
\[
18 \times 4 = 72
\]
\[
9 \times 5 = 45
\]
\[
72 \times 45 = 3240
\]
4. **Take the square root:**
\[
A = \sqrt{3240}
\]
To simplify \(\sqrt{3240}\):
\[
3240 = 36 \times 90 = 36 \times 9 \times 10 = 6^2 \times 3^2 \times 10
\]
Thus,
\[
\sqrt{3240} = 6 \times 3 \times \sqrt{10} = 18 \sqrt{10}
\]
5. **Approximate the area:**
The approximate value of \( \sqrt{10} \) is about \( 3.16 \), so:
\[
A \approx 18 \times 3.16 \approx 56.88 \text{ square inches}
\]
Thus, the area of the triangle is:
\[
\boxed{18 \sqrt{10}} \text{ square inches} \quad (\text{approximately } 56.88 \text{ square inches})
\]
1. **Calculate the semi-perimeter (s):**
\[
s = \frac{a + b + c}{2} = \frac{14 + 9 + 13}{2} = \frac{36}{2} = 18 \text{ in}
\]
2. **Apply Heron's formula:**
The area \(A\) of the triangle can be calculated using the formula:
\[
A = \sqrt{s(s - a)(s - b)(s - c)}
\]
Substitute \(s\) and the side lengths into the formula:
\[
A = \sqrt{18(18 - 14)(18 - 9)(18 - 13)}
\]
\[
A = \sqrt{18 \times 4 \times 9 \times 5}
\]
3. **Calculate the products:**
\[
18 \times 4 = 72
\]
\[
9 \times 5 = 45
\]
\[
72 \times 45 = 3240
\]
4. **Take the square root:**
\[
A = \sqrt{3240}
\]
To simplify \(\sqrt{3240}\):
\[
3240 = 36 \times 90 = 36 \times 9 \times 10 = 6^2 \times 3^2 \times 10
\]
Thus,
\[
\sqrt{3240} = 6 \times 3 \times \sqrt{10} = 18 \sqrt{10}
\]
5. **Approximate the area:**
The approximate value of \( \sqrt{10} \) is about \( 3.16 \), so:
\[
A \approx 18 \times 3.16 \approx 56.88 \text{ square inches}
\]
Thus, the area of the triangle is:
\[
\boxed{18 \sqrt{10}} \text{ square inches} \quad (\text{approximately } 56.88 \text{ square inches})
\]
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