Question

the triangle below is equilateral. find the length of side x in simplest radical form with a rational denominator. 11

Explanation:

Step1: Recall equilateral triangle properties

In an equilateral triangle, all sides are equal, and the altitude (the perpendicular from a vertex to the opposite side) bisects the base. So the base of the right triangle formed (half of the equilateral triangle's side) is $11$, and the hypotenuse is the side of the equilateral triangle, which is $x$. Wait, no—actually, in the equilateral triangle, when we draw the altitude, it splits the equilateral triangle into two $30$ - $60$ - $90$ right triangles. In a $30$ - $60$ - $90$ triangle, the sides are in the ratio $1:\sqrt{3}:2$, where the side opposite $30^{\circ}$ is the shortest one (let's call it $a$), the side opposite $60^{\circ}$ is $a\sqrt{3}$, and the hypotenuse is $2a$. But wait, in our case, the base of the right triangle (the side adjacent to the $30^{\circ}$ angle? Wait, no. Wait, the equilateral triangle has all angles $60^{\circ}$. When we draw the altitude, it creates two right triangles where the angle at the vertex is $30^{\circ}$? Wait, no. Wait, the original equilateral triangle has side length $x$, so each side is $x$. The altitude splits the base into two equal parts, so each part is $\frac{x}{2}$. But in the diagram, the length of the segment adjacent to the right angle (the base of the right triangle) is $11$. Wait, maybe I misread. Wait, the diagram shows that the base of the right triangle (the segment with length $11$) is half of the side of the equilateral triangle? Wait, no—wait, in an equilateral triangle, if we drop a perpendicular from a vertex to the opposite side, it bisects the opposite side. So if the length of the segment (the base of the right triangle) is $11$, then the full side of the equilateral triangle (the base) is $2\times11 = 22$. And since all sides of an equilateral triangle are equal, the side $x$ (which is a side of the equilateral triangle) is equal to the base, so $x = 22$. Wait, maybe I confused the right triangle sides. Wait, let's re - express. Let the equilateral triangle have side length $x$. The altitude divides the base into two segments of length $\frac{x}{2}$. But in the diagram, the length of the segment (the one with length $11$) is $\frac{x}{2}$? No, wait, maybe the $11$ is half of the side? Wait, no—wait, the right triangle has one leg as $11$, and the hypotenuse is $x$, and the other leg is the altitude. But in an equilateral triangle, all sides are equal. Wait, maybe the diagram is such that the segment of length $11$ is half of the side of the equilateral triangle. Wait, no, that can't be. Wait, no—wait, in the equilateral triangle, when you draw the altitude, the two right triangles formed have legs: one leg is the altitude $h$, the other leg is $\frac{x}{2}$ (half of the side of the equilateral triangle), and the hypotenuse is $x$ (the side of the equilateral triangle). But in the diagram, the length of the leg (the one with length $11$) is $\frac{x}{2}$? No, wait, maybe the $11$ is the length of the leg, and we need to find the hypotenuse. Wait, no—wait, the problem says "the triangle below is equilateral. Find the length of side $x$". So $x$ is a side of the equilateral triangle. The right triangle is formed by the altitude, so the base of the right triangle (the segment with length $11$) is half of the side of the equilateral triangle. Wait, that would mean that $\frac{x}{2}=11$, so $x = 22$. Yes, that makes sense. Because in an equilateral triangle, the altitude bisects the base, so if the length of the segment (the base of the right triangle) is $11$, then the full side of the equilateral triangle (the base) is $2\times11 = 22$, and since all sides of an equilateral triangle are equal, $x = 22$.

Answer:

$22$