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# Geometry

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Rhombus ABCD has perimeter 148, and one of its diagonals has length $$28$$. what is the area of ABCD?

Jan 20, 2022

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Because ABCD is a rhombus, then AB = BC = CD = AD.

AB = BC = CD = AD = 37 units.

One of rhombus ABCD's diagonals has length 28. We can divide 28 by 2 to get 14, which is the leg of a right triangle when rhombus ABCD is split into 4 congruent triangles.

The hypotenuse is 37, and the leg is 14, so that means the other leg is:$$\sqrt{37^2 - 14^2}$$  = $$\sqrt{1173}$$. Ew that looks ugly, but it is correct. That means the other diagonal has length $$2\sqrt{1173}$$

Since the area of a rhombus is one diagonal length times the other diagonal length over 2, then the area of this rhombus would be $$28\sqrt{1173}$$ units squared.

Jan 23, 2022