Question

question
the triangle below is equilateral. find the length of side x in simplest radical form with a rational denominator.
answer
attempt 1 out of 2
x =
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Explanation:

Step1: Use the property of equilateral triangle

In an equilateral triangle, the altitude divides the triangle into two \(30 - 60-90\) right - triangles. The side lengths of a \(30 - 60 - 90\) right - triangle are in the ratio \(1:\sqrt{3}:2\). Let the side length of the equilateral triangle be \(x\). The length of the shorter leg (half of the side of the equilateral triangle) is \(\frac{x}{2}\), and the length of the longer leg (altitude) is \(\frac{\sqrt{3}x}{2}\). But we can also use the Pythagorean theorem. If we consider the right - triangle formed by half of the side of the equilateral triangle (\(a=\frac{x}{2}\)), the altitude (\(b = 5\)) and the side of the equilateral triangle (\(c=x\)).

Step2: Apply the Pythagorean theorem

By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). Substitute \(a=\frac{x}{2}\), \(b = 5\) and \(c=x\) into the formula: \((\frac{x}{2})^{2}+5^{2}=x^{2}\).
Expand the equation: \(\frac{x^{2}}{4}+25=x^{2}\).
Move the terms involving \(x^{2}\) to one side: \(x^{2}-\frac{x^{2}}{4}=25\).
Combine like terms: \(\frac{4x^{2}-x^{2}}{4}=25\), so \(\frac{3x^{2}}{4}=25\).
Multiply both sides by \(\frac{4}{3}\): \(x^{2}=\frac{100}{3}\).
Take the square root of both sides: \(x=\sqrt{\frac{100}{3}}=\frac{10}{\sqrt{3}}\).
Rationalize the denominator: \(x=\frac{10\sqrt{3}}{3}\).

Answer:

\(\frac{10\sqrt{3}}{3}\)