The triangle below is equilateral. Find the length of side $x$ in simplest radical form with a ational denominator.

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Q. The triangle below is equilateral. Find the length of side

x

in simplest radical form with a ational denominator.

Equilateral Triangle Definition: An equilateral triangle has all sides equal and all angles equal to $60^\circ$ .
Perpendicular Bisects Side: If we drop a perpendicular from one vertex to the opposite side, it will bisect the side and form two $30$ - $60$ - $90$ right triangles.
$30$ $-60$ $-90$ Right Triangle Properties: In a $30$ - $60$ - $90$ triangle, the length of the side opposite the $30$ -degree angle is half the length of the hypotenuse.
Length of Side Opposite $30$ -degree Angle: Let's call the length of the side opposite the $30$ -degree angle $a$ . Then the hypotenuse, which is a side of the equilateral triangle, is $2a$ .
Length of Side Opposite $60$ -degree Angle: The length of the side opposite the $60$ -degree angle is $a\sqrt{3}$ , according to the properties of a $30$ - $60$ - $90$ triangle.
Length of Side $x$ in Equilateral Triangle: Since the perpendicular bisected the side of the equilateral triangle, the length of half the side is $a$ , and the full side is $2a$ .
Length of Side $x$ in Equilateral Triangle: Since the perpendicular bisected the side of the equilateral triangle, the length of half the side is $a$ , and the full side is $2a$ .Therefore, the length of side $x$ of the equilateral triangle is $2a$ .
Length of Side $x$ in Equilateral Triangle: Since the perpendicular bisected the side of the equilateral triangle, the length of half the side is $a$ , and the full side is $2a$ .Therefore, the length of side $x$ of the equilateral triangle is $2a$ .But we need to express $a$ in terms of $x$ . Since $x = 2a$ , then $a = \frac{x}{2}$ .
Length of Side $x$ in Equilateral Triangle: Since the perpendicular bisected the side of the equilateral triangle, the length of half the side is $a$ , and the full side is $2a$ .Therefore, the length of side $x$ of the equilateral triangle is $2a$ .But we need to express $a$ in terms of $x$ . Since $x = 2a$ , then $a = \frac{x}{2}$ .Substitute $a = \frac{x}{2}$ into the length of the side opposite the $60$ -degree angle, which is $a$ $0$ . We get $a$ $1$ .
Length of Side $x$ in Equilateral Triangle: Since the perpendicular bisected the side of the equilateral triangle, the length of half the side is $a$ , and the full side is $2a$ .Therefore, the length of side $x$ of the equilateral triangle is $2a$ .But we need to express $a$ in terms of $x$ . Since $x = 2a$ , then $a = \frac{x}{2}$ .Substitute $a = \frac{x}{2}$ into the length of the side opposite the $60$ -degree angle, which is $a$ $0$ . We get $a$ $1$ .Simplify $a$ $1$ to get $a$ $3$ . This is the length of side $x$ in simplest radical form with a rational denominator.