Finding the vertical height of a roof

Math Central

Quandaries & Queries

Question from Zainab, a student:

The question is: If the vertical height if a triangle is half the width of the base and the
slant length is 6 metres, find the exact vertical height of this part of the
roof.

I'm actually confused about finding out the height of an equilateral triangle
if you're only given the length or slant height. Please help! O.o

Hi Zainab.

Have you been learning about the Pythagorean theorem? That's how to
solve this problem.

Remember that it says that a² + b² = c², where c is the length of
the hypotenuse of a right triangle and a and b are the lengths of the
"legs" (the other two sides). What you are describing is a right
triangle, and the slant is the hypotenuse. So c = 6.

Now it sounds like you don't know a or b, so one equation isn't going
to be enough to solve the problem. But actually there is a second
equation given by the question: 2a = b, because the base is twice the
height. That means you can substitute 2a in for b in the Pythagorean
formula and you get this: a² + (2a)² = 6².

Now you should be able to solve the rest of the problem.

Hope this helps,
Stephen La Rocque.

P.S.: I'm not sure why you said "equilateral triangle", because that's
not implied by the question. If that's a separate question, you'll
need to send it to us separately.

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