Question

find the area of the triangle if each side is 10 inches.
draw
10
10
10
show your work here.

Explanation:

Step1: Identify the triangle type

The triangle has all sides equal (10 inches each), 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 by drawing a height, the base of each right triangle is \( \frac{s}{2}=\frac{10}{2} = 5 \) inches, and the hypotenuse is \( s = 10 \) inches.

Step2: Calculate the height

Using the Pythagorean theorem \( a^{2}+b^{2}=c^{2} \), where \( c = 10 \), \( a = 5 \), and \( b=h \) (height). So \( h=\sqrt{10^{2}-5^{2}}=\sqrt{100 - 25}=\sqrt{75}=5\sqrt{3} \) inches.

Step3: Calculate the area

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

Answer:

The area of the equilateral triangle is \( 25\sqrt{3} \) square inches (or approximately \( 43.3 \) square inches).