Question

the triangle below is equilateral. find the length of side x in simplest radical form with a rational denominator.

Explanation:

Step1: Recall equilateral - triangle property

In an equilateral triangle, the altitude divides the base into two equal parts and forms two right - triangles. Let the side length of the equilateral triangle be \(s\). Here, the hypotenuse of the right - triangle is \(s = 11\), and the base of the right - triangle is \(x\), and the altitude is the perpendicular from a vertex to the opposite side.

Step2: Apply the Pythagorean theorem

In a right - triangle, if the hypotenuse is \(c\) and the legs are \(a\) and \(b\), then \(a^{2}+b^{2}=c^{2}\). In our right - triangle formed from the equilateral triangle, \(c = 11\), and if the base is \(x\) and the other leg (altitude) is \(h\), and since the altitude of an equilateral triangle divides the base into two equal parts, we know that for the right - triangle, \(c = 11\), and the base is \(x\) and the other leg is \(\sqrt{11^{2}-x^{2}}\). Also, since the triangle is equilateral, if the side length of the equilateral triangle is \(s = 11\), and the base of the right - triangle formed by the altitude is \(x\), we have \(x=\frac{s}{2}\) (by the property of the altitude of an equilateral triangle). But we can also use the Pythagorean theorem: Let the side of the equilateral triangle be \(s = 11\), and if the base of the right - triangle is \(x\) and the hypotenuse is \(s\), then \(x^{2}+h^{2}=s^{2}\). Since the altitude of an equilateral triangle divides the base into two equal parts, we know that \(s = 11\), and if we consider the right - triangle formed, \(x=\frac{s}{2}\) (because of the symmetry of the equilateral triangle). In a right - triangle with hypotenuse \(s = 11\) and one leg \(x\) and the other leg \(h\), by the Pythagorean theorem \(x^{2}+h^{2}=11^{2}\). Also, since the altitude of an equilateral triangle divides the base into two equal parts, we know that if the side of the equilateral triangle is \(s\), then in the right - triangle formed, if the hypotenuse is \(s\) and the base is \(x\), we have \(x=\frac{s}{2}\). In this case, \(x=\frac{11}{2}\) (this is incorrect. Let's start over).
Let the side of the equilateral triangle be \(s\). The altitude of an equilateral triangle with side length \(s\) divides the triangle into two right - triangles. The hypotenuse of each right - triangle is \(s\), and the base of each right - triangle is \(\frac{s}{2}\).
We know that in a right - triangle with hypotenuse \(s = 11\) and base \(x\) (where \(x\) is half of the side of the equilateral triangle), using the Pythagorean theorem \(x^{2}+h^{2}=s^{2}\). Since the altitude \(h\) of an equilateral triangle with side \(s\) and base \(x=\frac{s}{2}\) gives us:
\[x^{2}+h^{2}=s^{2}\], and \(h=\sqrt{s^{2}-x^{2}}\). In our case, \(s = 11\), and the right - triangle has hypotenuse \(s = 11\) and base \(x\).
By the Pythagorean theorem, if the hypotenuse \(c = 11\) and one leg is \(x\) and the other leg is \(y\), \(x^{2}+y^{2}=121\). In an equilateral triangle, if the side length is \(11\), and we consider the right - triangle formed by the altitude, the base of the right - triangle \(x\) satisfies \(x^{2}+(\sqrt{11^{2}-x^{2}})^{2}=11^{2}\). Since the altitude of an equilateral triangle divides the base into two equal parts, if the side of the equilateral triangle is \(s = 11\), the base of the right - triangle \(x\) is related to the hypotenuse \(s\) by \(x=\frac{s}{2}\) (not correct way).
Let the side of the equilateral triangle be \(s = 11\). The right - triangle formed by the altitude has hypotenuse \(s = 11\) and base \(x\). By the Pythagorean theorem \(x^{2}+h^{2}=s^{2}\), where \(h\) is the altitude. In an equilateral triangle, if the side length is \(s\), the altitude \(h=\frac{\sqrt{3}}{2}s\) and the base of the right - triangle (half of the side of the equilateral triangle) is \(x\).
We know that in the right - triangle, \(x^{2}+(\frac{\sqrt{3}}{2}s)^{2}=s^{2}\). Substituting \(s = 11\), we have \(x^{2}+(\frac{\sqrt{3}}{2}\times11)^{2}=11^{2}\).
\[x^{2}=11^{2}-\frac{3}{4}\times11^{2}=\frac{1}{4}\times11^{2}\]
\[x=\frac{11}{2}\]

Answer:

\(\frac{11}{2}\)