The altitude of equilateral triangle ABC is 15. What is the length of a side of triangle ABC?
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To find the length of a side of the equilateral triangle ABC, we can use the formula for the altitude of an equilateral triangle. The altitude divides the triangle into two congruent right triangles. Let's call the length of a side of the triangle x.
In an equilateral triangle, the altitude is also the perpendicular bisector of the base. This divides the base into two segments of equal length, forming a right triangle with a hypotenuse equal to x (the side length), and a base half the length of the entire base, which is x/2.
Now, we can use the Pythagorean theorem to solve for x. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Using this information, we have:
x^2 = (x/2)^2 + 15^2
Simplifying:
x^2 = x^2/4 + 225
Multiplying both sides by 4 to eliminate the fraction:
4x^2 = x^2 + 900
Moving all terms to one side:
3x^2 = 900
Dividing both sides by 3:
x^2 = 300
Taking the square root of both sides:
x = √300
Simplifying further:
x ≈ 17.32
Therefore, the length of a side of triangle ABC is approximately 17.32 units.