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Answer :
To find the area of a triangle when its side lengths are given, you can use Heron's formula. Here’s how you can do it step-by-step:
1. Identify the sides of the triangle:
The side lengths given are:
- [tex]\( a = 25 \)[/tex]
- [tex]\( b = 13 \)[/tex]
- [tex]\( c = 17 \)[/tex]
2. Calculate the semi-perimeter of the triangle:
The semi-perimeter, denoted as [tex]\( s \)[/tex], is calculated as half the perimeter of the triangle:
[tex]\[
s = \frac{a + b + c}{2} = \frac{25 + 13 + 17}{2} = 27.5
\][/tex]
3. Apply Heron's formula to find the area:
Heron's formula states that the area of a triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] is:
[tex]\[
\text{Area} = \sqrt{s \cdot (s - a) \cdot (s - b) \cdot (s - c)}
\][/tex]
Plug in the values for [tex]\( s \)[/tex], [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\text{Area} = \sqrt{27.5 \cdot (27.5 - 25) \cdot (27.5 - 13) \cdot (27.5 - 17)}
\][/tex]
Simplify the expression:
[tex]\[
\text{Area} = \sqrt{27.5 \cdot 2.5 \cdot 14.5 \cdot 10.5}
\][/tex]
4. Calculate the result:
When you calculate this, the area comes out to approximately 102.31 square units.
5. Choose the closest option from the given choices:
The options given are:
- a. 99.1 units[tex]\(^2\)[/tex]
- b. 100.5 units[tex]\(^2\)[/tex]
- c. 98.7 units[tex]\(^2\)[/tex]
- d. 102.3 units[tex]\(^2\)[/tex]
The area calculated as 102.31 is closest to option d. 102.3 units[tex]\(^2\)[/tex].
Therefore, the best answer is d. 102.3 units[tex]\(^2\)[/tex].
1. Identify the sides of the triangle:
The side lengths given are:
- [tex]\( a = 25 \)[/tex]
- [tex]\( b = 13 \)[/tex]
- [tex]\( c = 17 \)[/tex]
2. Calculate the semi-perimeter of the triangle:
The semi-perimeter, denoted as [tex]\( s \)[/tex], is calculated as half the perimeter of the triangle:
[tex]\[
s = \frac{a + b + c}{2} = \frac{25 + 13 + 17}{2} = 27.5
\][/tex]
3. Apply Heron's formula to find the area:
Heron's formula states that the area of a triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] is:
[tex]\[
\text{Area} = \sqrt{s \cdot (s - a) \cdot (s - b) \cdot (s - c)}
\][/tex]
Plug in the values for [tex]\( s \)[/tex], [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\text{Area} = \sqrt{27.5 \cdot (27.5 - 25) \cdot (27.5 - 13) \cdot (27.5 - 17)}
\][/tex]
Simplify the expression:
[tex]\[
\text{Area} = \sqrt{27.5 \cdot 2.5 \cdot 14.5 \cdot 10.5}
\][/tex]
4. Calculate the result:
When you calculate this, the area comes out to approximately 102.31 square units.
5. Choose the closest option from the given choices:
The options given are:
- a. 99.1 units[tex]\(^2\)[/tex]
- b. 100.5 units[tex]\(^2\)[/tex]
- c. 98.7 units[tex]\(^2\)[/tex]
- d. 102.3 units[tex]\(^2\)[/tex]
The area calculated as 102.31 is closest to option d. 102.3 units[tex]\(^2\)[/tex].
Therefore, the best answer is d. 102.3 units[tex]\(^2\)[/tex].
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