- "Chat" with the following literal using GPT 4 Turbo.
how would I calculate the height of an equilateral triangle assuming the edge is x
The text:
To calculate the height of an equilateral triangle when the length of each side is given as $x$, you can use the Pythagorean theorem. The height of an equilateral triangle forms a right triangle with half of the base and the full height as the sides.
Here are the steps to find the height:
1. **Divide the base into two equal parts**: Since the triangle is equilateral, you can split one side into two equal parts, each of length $\frac{x}{2}$.
2. **Apply the Pythagorean theorem**: The height ($h$) of the triangle, the half base ($\frac{x}{2}$), and the full side ($x$) form a right triangle. According to the Pythagorean theorem:
$$
x^2 = h^2 + \left(\frac{x}{2}\right)^2
$$
3. **Solve for $h$**:
$$
x^2 = h^2 + \left(\frac{x}{2}\right)^2 = h^2 + \frac{x^2}{4}
$$
Rearrange to solve for $h^2$:
$$
h^2 = x^2 - \frac{x^2}{4} = \frac{4x^2}{4} - \frac{x^2}{4} = \frac{3x^2}{4}
$$
Taking the square root of both sides, we get:
$$
h = \sqrt{\frac{3x^2}{4}} = \frac{x\sqrt{3}}{2}
$$
Therefore, the height $h$ of an equilateral triangle with side length $x$ is $\frac{x\sqrt{3}}{2}$.
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