Math CentralQuandaries & Queries


Question from phoenix, a student:

given triangle with angles 30, 105, 45 degrees. base = 2radical 2, no height given, find the area


Hi Phoenix,

You didn't say which side is the base. If the 105o angle is the apex then the diagram is approximately


AD is perpendicular to BC so |AD| = h is the height of the triangle. I let |DC| = x and then |BD| = 2√2 - x. You can use a trig function to relate h and x in two ways, one by using the 45o angle and the second by using the 30o angle. This gives you two linear equations in h and x which you can solve for h.

If the 105o angle is not the apex then D will be on an extension of the base. In this case let x be the distance from D to the nearest vertex on the base and the the distance from D to the furthest vertex will be 2√2 + x. A similar use of trig functions will produce equations that can be solved for h.


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