GMAT Quantitative - Calculating the length of the side of an equilateral triangle

If the area of an equilateral is , given a height of , what is the base of the triangle?

Explanation

We derive the equation of base of a triangle from the area of a triangle formula:

$A=\frac{bh}{2}$

$b=\frac{2A}{h}$

$b=\frac{2*12}{5}$

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Equilateral triangle is inscribed in a circle with radius , what is the length of a side of the triangle?

$\frac{15}{\sqrt{3}}$

$3\sqrt{5}$

$5\sqrt{3}$

$\sqrt{5}$

$\frac{15}{2}$

Explanation

Since we are given the radius, we should be able to find the height of the equilateral triangle. Indeed, the center of the circle is at the intersections of the heights of the triangle, and is located $\frac{2}{3}$ away from the edge of a given height.

Therefore 5, the radius of the circle is $\frac{2}{3}$ of the height.

Therefore, the height must be $\frac{15}{2}$ .

From here, we can use the formula for the height of the equilateral triangle $h=s\frac{\sqrt{3}}{2}$ , where is the length of the height and is the length of a side of the equilateral triangle.