Trapezoid question, with focus on triangle in problem.

Question

Hi Everyone,

Please see image. Can someone explain to me why I cannot use the 2 to find the sides of the triangle in the trapezoid? My thought process is either to assume the triangle is 30-60-90, or use the Pythagorean theorem. What rule restricts me from doing either of the two?

Would love some help! Thanks.

Edit: Understand why not Pythagorean, but why can I not assume it is 30-60-90 given shortest side, and right angle?

IanStewart · Answer

Among the more common mistakes test takers make in Geometry is "seeing 30-60-90 triangles" when they aren't really there. You can never assume a triangle is a 30-60-90. You need to prove it is, either by finding the angles in the triangle and seeing they are specifically 30, 60 and 90 degrees, or by noticing the side lengths are in a 1 to √3 to 2 ratio. If you can't show that one of those two things is true, you'll almost always be making a mistake if you assume the triangle is a 30-60-90 (or if you're not making a mistake, you've just made a very lucky guess).

It would be helpful to see the original question here, but I assume it was a DS question that asked something like:

What is the area of the trapezoid above (with the diagram)?
1. The height is 1/8 of the trapezoid's area
2. something else

Before using either Statement, you can imagine drawing the diagram in different ways -- if the height is very small, say 0.0001, angle x will be very small, very close to 0 degrees. If the height is enormous, say 1,000,000, then x will be very big, nearly 90 degrees. So x can be anything between 0 and 90 here, and we can't assume it's either 30 or 60.

And if Statement 1 reads as I've written it above, it doesn't give any new information, incidentally. To find the area of a trapezoid, you multiply the average of the two unequal sides by the height. The average of the two unequal sides here is 8 (one is 10, the other is 6). So the area is 8h, and if A = 8h, then h = (1/8)*A. So Statement 1, as I've written it, is something we already knew just from facts about trapezoids and is not at all useful. But I'm just guessing what the question might have said.

Trapezoid question, with focus on triangle in problem.

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