Answer :
To find the area of a triangle with side lengths [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 [tex]$A$[/tex] is given by
[tex]$$
A = \sqrt{s(s-a)(s-b)(s-c)},
$$[/tex]
where the semiperimeter [tex]$s$[/tex] is
[tex]$$
s = \frac{a+b+c}{2}.
$$[/tex]
Step 1. Compute the semiperimeter
First, calculate [tex]$s$[/tex]:
[tex]$$
s = \frac{25 + 13 + 17}{2} = \frac{55}{2} = 27.5.
$$[/tex]
Step 2. Calculate the differences
Now, compute each of the expressions [tex]$(s-a)$[/tex], [tex]$(s-b)$[/tex], and [tex]$(s-c)$[/tex]:
[tex]$$
s - a = 27.5 - 25 = 2.5,
$$[/tex]
[tex]$$
s - b = 27.5 - 13 = 14.5,
$$[/tex]
[tex]$$
s - c = 27.5 - 17 = 10.5.
$$[/tex]
Step 3. Evaluate the product in Heron’s formula
Multiply these values along with [tex]$s$[/tex]:
[tex]$$
\text{Product} = s(s-a)(s-b)(s-c) = 27.5 \times 2.5 \times 14.5 \times 10.5.
$$[/tex]
The multiplication yields
[tex]$$
27.5 \times 2.5 \times 14.5 \times 10.5 \approx 10467.1875.
$$[/tex]
Step 4. Compute the area
Take the square root of the product to find the area:
[tex]$$
A = \sqrt{10467.1875} \approx 102.3.
$$[/tex]
Thus, the area of the triangle is approximately [tex]$102.3$[/tex] units[tex]$^2$[/tex].
Final Answer: Option D is the correct choice.
[tex]$$
A = \sqrt{s(s-a)(s-b)(s-c)},
$$[/tex]
where the semiperimeter [tex]$s$[/tex] is
[tex]$$
s = \frac{a+b+c}{2}.
$$[/tex]
Step 1. Compute the semiperimeter
First, calculate [tex]$s$[/tex]:
[tex]$$
s = \frac{25 + 13 + 17}{2} = \frac{55}{2} = 27.5.
$$[/tex]
Step 2. Calculate the differences
Now, compute each of the expressions [tex]$(s-a)$[/tex], [tex]$(s-b)$[/tex], and [tex]$(s-c)$[/tex]:
[tex]$$
s - a = 27.5 - 25 = 2.5,
$$[/tex]
[tex]$$
s - b = 27.5 - 13 = 14.5,
$$[/tex]
[tex]$$
s - c = 27.5 - 17 = 10.5.
$$[/tex]
Step 3. Evaluate the product in Heron’s formula
Multiply these values along with [tex]$s$[/tex]:
[tex]$$
\text{Product} = s(s-a)(s-b)(s-c) = 27.5 \times 2.5 \times 14.5 \times 10.5.
$$[/tex]
The multiplication yields
[tex]$$
27.5 \times 2.5 \times 14.5 \times 10.5 \approx 10467.1875.
$$[/tex]
Step 4. Compute the area
Take the square root of the product to find the area:
[tex]$$
A = \sqrt{10467.1875} \approx 102.3.
$$[/tex]
Thus, the area of the triangle is approximately [tex]$102.3$[/tex] units[tex]$^2$[/tex].
Final Answer: Option D is the correct choice.
