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Geometry Level 4

What is the largest possible area of a rectangle(in square units) inscribed in the triangle shown in the picture above?


The answer is 42.

Shiv Gaur by Shiv Gaur , 50, Kazakhstan

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2 solutions

Gacon Noname
May 27, 2014

BC = a, AH = h: a = 21; h = 8. x/a + y/h =1 we also have rectangular area = xy so using Cauchy, we have: Max xy = 42

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Why must the rectangle be built on the base of side 21? How do you know that a tilted rectangle can't have a larger area?

Calvin Lin Staff - 7 years ago

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actually, we can choose any segment of triangle, because a.h = 2S ( S is area of the triangle).

Gacon Noname - 7 years ago

The longest side, opposite the biggest angle, must be one side of the rectangle, by simple geometric reasoning. Solve for angles using Cosine rule. Then formulate area as a function of height (h is side of the recangle perpendicular to the longest side, differentiate expression to find maximum and solve for h. h = 4 if you do this right. and th ebase lenght is 10.5, (21/2). A geometrical way of looking at it, that doesn't need calculus, is that if you make a parallelogram with sides 21 and 10, say, then the area of the parallelogram can be found (I still need to use the cosine rule to get the perpendicular from the obtuse andlge to the side 21). This perpendicular is 8, so the area of the parallelogram is 21 8=168 is the area of a 21 8 rectangle. Cutting this rectangle with a diagonal, its obvious from symmetry that hte maximum area rectangle in one of the triangles is 1/4 of the area of the parent rectangle = 1/4 of the area of the parent parallelogram, which takes us back to our original 10 by 17 by 21 triangle. If you follow. So 168/4 = 42. Also, of course, the easy solution is that 42 is the answer to life, the univers, and everything. The triangle is something, that is contained in everything, so the answer is 42. QED

Andrew Rakich - 6 years, 10 months ago

Since the biggest rectangle has an area half the area of the triangle, what ever side we take,
the answer is the same.

If the rectangle is tilted the proportional increase in on side is less than proportional decrease on the other side.
A right triangle will have this decrease the lest. Hence if it is proved for a right triangle, it will be true for all triangles.
Consider triangle ABD. If E is moved up or down, area decreases.
Let E be as it is. But H moved to the right to H'. Angle between EH and EH' is say X. So the increase is 1/CosX. Now H'D' = (HD-HH')/CosX. This is prepositionally more decrease than increase in EH. Hence the area reduces. If D was not right angle the, H'D' would be further shortened.
For obtuse angle triangle, rectangle on shorter side would not touch all three sides.


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