# The altitude of an equilateral triangle is 12. What is the length of a side and what is the area of the triangle?

Length of one side is

Let side length, altitude (height), and area be s, h, and A respectively.

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The length of a side of the equilateral triangle can be found using the formula ( \text{side length} = \frac{2 \times \text{altitude}}{\sqrt{3}} ), where the altitude is given as 12.

Substitute the altitude into the formula: [ \text{side length} = \frac{2 \times 12}{\sqrt{3}} ] [ \text{side length} = \frac{24}{\sqrt{3}} ] [ \text{side length} = \frac{24 \times \sqrt{3}}{3} ] [ \text{side length} = 8\sqrt{3} ]

The area of an equilateral triangle can be calculated using the formula ( \text{area} = \frac{\sqrt{3}}{4} \times \text{side length}^2 ).

Substitute the side length into the area formula: [ \text{area} = \frac{\sqrt{3}}{4} \times (8\sqrt{3})^2 ] [ \text{area} = \frac{\sqrt{3}}{4} \times 192 ] [ \text{area} = \frac{192\sqrt{3}}{4} ] [ \text{area} = 48\sqrt{3} ]

So, the length of a side of the equilateral triangle is ( 8\sqrt{3} ) units, and the area of the triangle is ( 48\sqrt{3} ) square units.

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