Question

find the area of the triangle if each side is 10 centimeters.

Explanation:

Step1: Identify the triangle type

The triangle has all sides equal (each 10 cm), so it's an equilateral triangle. For an equilateral triangle with side length \( s \), we can find the height \( h \) using the Pythagorean theorem. If we split the equilateral triangle into two right triangles, the base of each right triangle is \( \frac{s}{2}=\frac{10}{2} = 5 \) cm, the hypotenuse is \( s = 10 \) cm, and the height \( h \) is the other leg.
Using \( a^{2}+b^{2}=c^{2} \), where \( c = 10 \), \( a = 5 \), we solve for \( b \) (height \( h \)):
\( h^{2}+5^{2}=10^{2} \)
\( h^{2}=100 - 25=75 \)
\( h=\sqrt{75}=5\sqrt{3} \) cm.

Step2: Calculate the area

The formula for the area of a triangle is \( A=\frac{1}{2}\times base\times height \). For the equilateral triangle, base \( = 10 \) cm, height \( = 5\sqrt{3} \) cm.
So \( A=\frac{1}{2}\times10\times5\sqrt{3} \)
\( A = 5\times5\sqrt{3}=25\sqrt{3}\approx25\times1.732 = 43.3 \) square centimeters.

Answer:

The area of the equilateral triangle is \( 25\sqrt{3}\approx43.3 \) square centimeters (or exactly \( 25\sqrt{3} \) \( \text{cm}^2 \)).