College

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ 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

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].