To find the area of a triangle when the lengths of all three sides are given, we can use Heron's formula. Here are the steps to compute the area:
1. Identify the sides of the triangle:
Let's denote the sides of the triangle as:
[tex]\[
a = 10, \quad b = 10, \quad c = 16
\][/tex]
2. Calculate the semi-perimeter (s):
The semi-perimeter of a triangle is given by:
[tex]\[
s = \frac{a + b + c}{2}
\][/tex]
Substituting the given values:
[tex]\[
s = \frac{10 + 10 + 16}{2} = \frac{36}{2} = 18
\][/tex]
3. Apply Heron's formula:
Heron's formula states that the area [tex]\( A \)[/tex] of a triangle can be calculated as:
[tex]\[
A = \sqrt{s \cdot (s - a) \cdot (s - b) \cdot (s - c)}
\][/tex]
Plugging in the values we have:
[tex]\[
A = \sqrt{18 \cdot (18 - 10) \cdot (18 - 10) \cdot (18 - 16)}
\][/tex]
Simplifying inside the square root:
[tex]\[
A = \sqrt{18 \cdot 8 \cdot 8 \cdot 2}
\][/tex]
Combining the terms inside the square root:
[tex]\[
A = \sqrt{18 \times 128}
\][/tex]
Calculating the product inside:
[tex]\[
A = \sqrt{2304}
\][/tex]
Taking the square root of 2304:
[tex]\[
A = 48
\][/tex]
4. Conclusion:
The area of the triangle with sides [tex]\(10 \times 10 \times 16\)[/tex] is [tex]\(48\)[/tex] square units. Therefore, the correct answer is:
[tex]\[
\boxed{48}
\][/tex]
