Question

A square and an equiangular triangle have the same
perimeter. If the area of the square is 144 cm2, what is
the length of one side of the triangle?

A.
24 cm
B.
16 cm
C.
12 cm
D.
8 cm
E.
4 cm

Answer 1

To solve this problem, we first need to find the side length of the square.

1. **Calculate the side length of the square:**
The area of the square is given as 144 cm². The area of a square is given by the formula:
\[
\text{Area} = \text{side}^2
\]
Therefore, we can find the side length by taking the square root of the area:
\[
\text{side} = \sqrt{144} = 12 \, \text{cm}
\]

2. **Calculate the perimeter of the square:**
The perimeter \( P \) of a square with side length \( s \) is given by:
\[
P = 4 \times \text{side} = 4 \times 12 = 48 \, \text{cm}
\]

3. **Determine the perimeter of the equiangular triangle:**
Since the square and the equiangular triangle have the same perimeter, the perimeter of the triangle is also 48 cm.

4. **Calculate the side length of the equiangular triangle:**
Let \( a \) be the length of one side of the equiangular triangle. The perimeter \( P \) of an equiangular triangle is given by:
\[
P = 3 \times a
\]
Setting this equal to the perimeter we found previously:
\[
3a = 48
\]
Solving for \( a \):
\[
a = \frac{48}{3} = 16 \, \text{cm}
\]

Thus, the length of one side of the triangle is \( 16 \, \text{cm} \).

The correct answer is:
**B. 16 cm**