Question

Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side L if one side of the rectangle lies on the base of the triangle.

base=
height=

Answer 1

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Answer:

• base: L/2
• height: L√3/2

Explanation:

Let x represent the ratio of the rectangle base to the triangle side length. Then the height of the small triangle above the rectangle will be x times the height of the equilateral triangle. Then the height of the rectangle is (1-x) times the height of the equilateral triangle. The rectangle's area will be ...

A = bh

A = (xL)(1-x)(L·√3/2) = (L²√3/2)(x)(1-x)

This graphs as parabola opening downward with x-intercepts at x=0 and x=1. The vertex is on the line of symmetry, halfway between these zeros, at x = 1/2.

The base of the rectangle is L/2.

The height of the rectangle is L√3/2.

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The general solution to this sort of problem is that one side of the rectangle is the midsegment of the triangle.