A tetrahedron is made of a base which is an equilateral triangle. The length of each slant edge of the tetrahedron is 20m while that of each side of its base is 30m. Calculate the... A tetrahedron is made of a base which is an equilateral triangle. The length of each slant edge of the tetrahedron is 20m while that of each side of its base is 30m. Calculate the shortest distance from the apex of the tetrahedron to its base. Read more

Steps to Solve

1. Find the centroid of the equilateral triangle

The shortest distance from the apex to the base lands at the centroid of the equilateral triangle. The centroid is the point where the medians of the triangle intersect, and it's also the center of the circumcircle. The distance from a vertex to the centroid is 2/3 of the length of the median.

2. Calculate the length of the median of the equilateral triangle

   In an equilateral triangle, the median is also the altitude. We can calculate its length using the Pythagorean theorem. If the side length of the equilateral triangle is $s$, then the median $m$ satisfies $$m^2 + (\frac{s}{2})^2 = s^2$$ $$m^2 = s^2 - \frac{s^2}{4} = \frac{3s^2}{4}$$ $$m = \frac{s\sqrt{3}}{2}$$ Given that $s = 30$ m, we have $$m = \frac{30\sqrt{3}}{2} = 15\sqrt{3} \text{ m}$$

3. Calculate the distance from a vertex to the centroid

   This distance, $d$, is $\frac{2}{3}$ of the median $m$. $$d = \frac{2}{3}m = \frac{2}{3}(15\sqrt{3}) = 10\sqrt{3} \text{ m}$$

4. Calculate the height of the tetrahedron

   Now consider the right triangle formed by the height $h$ of the tetrahedron, the slant edge (length 20 m), and the distance $d$ from a vertex to the centroid. Using the Pythagorean theorem: $$h^2 + d^2 = (\text{slant edge})^2$$ $$h^2 = 20^2 - (10\sqrt{3})^2$$ $$h^2 = 400 - 100(3) = 400 - 300 = 100$$ $$h = \sqrt{100} = 10 \text{ m}$$