Question

the height of the equilateral triangle shown divides the triangle into two congruent right triangles. what is the height, rounded to the nearest tenth? enter the answer. 8 8 8 about units

Explanation:

Step1: Identify the sides of the right triangle

The equilateral triangle has side length 8. When we draw the height, it splits the base into two equal parts, so the base of each right triangle is $\frac{8}{2} = 4$. The hypotenuse of each right triangle is the side of the equilateral triangle, which is 8. Let the height be $h$.

Step2: Apply the Pythagorean theorem

The Pythagorean theorem states that for a right triangle with legs $a$ and $b$ and hypotenuse $c$, $a^2 + b^2 = c^2$. Here, $a = h$, $b = 4$, and $c = 8$. So we have the equation $h^2 + 4^2 = 8^2$.

Step3: Solve for $h$

First, calculate $4^2 = 16$ and $8^2 = 64$. Then the equation becomes $h^2 + 16 = 64$. Subtract 16 from both sides: $h^2 = 64 - 16 = 48$. Take the square root of both sides: $h = \sqrt{48}$. Simplify $\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \approx 4 \times 1.732 = 6.928$.

Step4: Round to the nearest tenth

Rounding 6.928 to the nearest tenth gives 6.9.

Answer:

6.9