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I was going through some Olympiad Maths and found this question:

Given a trapezoid with its upper base $5$ cm long, lower base $10$ cm long, and its legs are $3$ and $4$ cm long. What is the area of this trapezoid?

Yeah, I know. There are equations to calculate this, I found some equations on Math Stack Exchange too.

What I don’t understand is that this is an Olympiad question. The proofs that I saw to create the formulae did not look like something that should appear in an Olympiad question. Am I missing something, or do I actually need to create my own formula to solve this question? Keep in mind that this is a timed test; if I was actually taking this test, I would have to solve this in 2 minutes maximum.

Blue
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Tyrcnex
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    Try to make use of a $3-4-5$ right triangle. – MyMolecules Nov 14 '21 at 06:02
  • Reading the Olympiad question, I assumed you wondered why it was an Olympiad problem because it is nearly trivial. But the last paragraph suggests that maybe you believe it is too difficult? Could you clarify? – Servaes Nov 14 '21 at 14:09
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    In general, questions which implore us to get into the minds of the authors of texts or exams are off-topic here. Such questions are not actually mathematics questions---we can't read other people's minds. As such the question "Why is this an Olympiad question?" is off-topic. If you are actually looking for a solution to the question posed, please edit the question to make it clear that this is the concern. – Xander Henderson Nov 14 '21 at 16:08

3 Answers3

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Draw a parallelogram with the upper base and one of the trapezoid's legs as two of the parallelogram's sides. Diagram of parallelogram and trapezoid Notice the right-angle triangle? You can use this to find the height of the trapezoid, and thus its area.

VTand
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We join the midpoint of longer base to endpoints of shorter base and find that the trapezium is partitioned into three $3-4-5$ right triangles.

Hence area is $3 \times 6=18$.

MyMolecules
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Well, it is easy to see it breaks into finding the areas of a rectangle and two right triangles, but stacking the right triangles together gives a $3-4-5$ triangle, whose area is easily measured. This area gives the height, which allows the rectangle to be measured easily as well. Shouldn't take much time I suppose.

Macavity
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