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Answer :
To find the area of the triangle with side lengths
[tex]$$a = 17,\quad b = 13,\quad c = 19,$$[/tex]
we can use Heron's formula. This formula states that the area of a triangle is given by
[tex]$$
\text{Area} = \sqrt{s (s-a) (s-b) (s-c)},
$$[/tex]
where the semi-perimeter [tex]$s$[/tex] is
[tex]$$
s = \frac{a + b + c}{2}.
$$[/tex]
Step 1. Calculate the semi-perimeter, [tex]$s$[/tex]:
[tex]$$
s = \frac{17 + 13 + 19}{2} = \frac{49}{2} = 24.5.
$$[/tex]
Step 2. Find the differences [tex]$s-a$[/tex], [tex]$s-b$[/tex], and [tex]$s-c$[/tex]:
[tex]$$
s - a = 24.5 - 17 = 7.5,
$$[/tex]
[tex]$$
s - b = 24.5 - 13 = 11.5,
$$[/tex]
[tex]$$
s - c = 24.5 - 19 = 5.5.
$$[/tex]
Step 3. Substitute the values into Heron’s formula:
[tex]$$
\text{Area} = \sqrt{24.5 \times 7.5 \times 11.5 \times 5.5}.
$$[/tex]
Step 4. Compute the product inside the square root:
[tex]$$
P = 24.5 \times 7.5 \times 11.5 \times 5.5.
$$[/tex]
After evaluating the product, we have
[tex]$$
P \approx 11622.1875.
$$[/tex]
Step 5. Take the square root to find the area:
[tex]$$
\text{Area} = \sqrt{11622.1875} \approx 107.8.
$$[/tex]
Thus, the area of the triangle is approximately [tex]$107.8$[/tex] square units.
Among the given choices, the best answer is
C. 107.8 units[tex]$^2$[/tex].
[tex]$$a = 17,\quad b = 13,\quad c = 19,$$[/tex]
we can use Heron's formula. This formula states that the area of a triangle is given by
[tex]$$
\text{Area} = \sqrt{s (s-a) (s-b) (s-c)},
$$[/tex]
where the semi-perimeter [tex]$s$[/tex] is
[tex]$$
s = \frac{a + b + c}{2}.
$$[/tex]
Step 1. Calculate the semi-perimeter, [tex]$s$[/tex]:
[tex]$$
s = \frac{17 + 13 + 19}{2} = \frac{49}{2} = 24.5.
$$[/tex]
Step 2. Find the differences [tex]$s-a$[/tex], [tex]$s-b$[/tex], and [tex]$s-c$[/tex]:
[tex]$$
s - a = 24.5 - 17 = 7.5,
$$[/tex]
[tex]$$
s - b = 24.5 - 13 = 11.5,
$$[/tex]
[tex]$$
s - c = 24.5 - 19 = 5.5.
$$[/tex]
Step 3. Substitute the values into Heron’s formula:
[tex]$$
\text{Area} = \sqrt{24.5 \times 7.5 \times 11.5 \times 5.5}.
$$[/tex]
Step 4. Compute the product inside the square root:
[tex]$$
P = 24.5 \times 7.5 \times 11.5 \times 5.5.
$$[/tex]
After evaluating the product, we have
[tex]$$
P \approx 11622.1875.
$$[/tex]
Step 5. Take the square root to find the area:
[tex]$$
\text{Area} = \sqrt{11622.1875} \approx 107.8.
$$[/tex]
Thus, the area of the triangle is approximately [tex]$107.8$[/tex] square units.
Among the given choices, the best answer is
C. 107.8 units[tex]$^2$[/tex].
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