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------------------------------------------------ Find the area of a triangle with sides [tex]a=17[/tex], [tex]b=13[/tex], and [tex]c=19[/tex].

a. [tex]99.7[/tex] units[tex]$^2$[/tex]
b. [tex]104.5[/tex] units[tex]$^2$[/tex]
c. [tex]107.8[/tex] units[tex]$^2$[/tex]
d. [tex]112.5[/tex] units[tex]$^2$[/tex]

Please select the best answer from the choices provided:
A
B
C
D

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].