### Step-by-Step Solution for Finding the Length of Equal Sides of an Isosceles Triangle
Given:
- The area of the isosceles triangle: [tex]\( 48 \, \text{cm}^2 \)[/tex]
- The base of the isosceles triangle: [tex]\( 12 \, \text{cm} \)[/tex]
To Find:
- The length of the equal sides (denoted as [tex]\( a \)[/tex])
Formulas:
1. Area of an isosceles triangle:
[tex]\[
\text{Area} = \frac{b}{4} \sqrt{4a^2 - b^2}
\][/tex]
where [tex]\( b \)[/tex] is the base and [tex]\( a \)[/tex] is the length of the equal sides.
2. Perimeter of an isosceles triangle:
[tex]\[
\text{Perimeter} = 2a + b
\][/tex]
Steps to Find the Length of the Equal Sides:
- Given the area formula for an isosceles triangle, start with:
[tex]\[
48 = \frac{12}{4} \sqrt{4a^2 - 12^2}
\][/tex]
- Simplify the fraction:
[tex]\[
48 = 3 \sqrt{4a^2 - 144}
\][/tex]
- Divide both sides by 3:
[tex]\[
16 = \sqrt{4a^2 - 144}
\][/tex]
- Square both sides to eliminate the square root:
[tex]\[
16^2 = 4a^2 - 144
\][/tex]
- Calculate [tex]\( 16^2 \)[/tex]:
[tex]\[
256 = 4a^2 - 144
\][/tex]
- Add 144 to both sides:
[tex]\[
256 + 144 = 4a^2
\][/tex]
- Combine like terms:
[tex]\[
400 = 4a^2
\][/tex]
- Divide by 4 to solve for [tex]\( a^2 \)[/tex]:
[tex]\[
100 = a^2
\][/tex]
- Take the square root of both sides:
[tex]\[
a = \sqrt{100}
\][/tex]
- Find the value of [tex]\( a \)[/tex]:
[tex]\[
a = 10
\][/tex]
Answer:
The length of the equal sides of the isosceles triangle is [tex]\( 10 \, \text{cm} \)[/tex].
