Verifying the Height of an Equilateral Triangle

In this problem, we're asked to determine if the line segment M Q is the height of the equilateral triangle L M N, given the side lengths shown in the diagram.

If M Q is indeed the height of an equilateral triangle, it must be perpendicular to the base. This means triangle L M Q would have to be a right-angled triangle.

We can verify this by checking if the lengths satisfy the Pythagorean theorem: a squared plus b squared equals c squared.

Let's list our known values from the diagram. The side L M is the square root of thirty.

Now, let's substitute these into the left side of our Pythagorean equation. We'll find the sum of the squares of L Q and M Q.

Squaring the first term, the square root of thirty squared is thirty, and two squared is four. So we get thirty over four.

Simplifying thirty over four gives us seven point five. Then, the square of the square root of fifteen is simply fifteen.