A Rhombus Overlapping a Square

Geometry Level 1

In the image bellow the rhombus P Q D C PQDC is overlapping the square A B C D ABCD . The square A B C D ABCD has an area of 25 cm 2 25\text{cm}^{2} and the rhombus P Q D C PQDC has an area of 20 cm 2 20\text{cm}^{2} . What is the gray area in cm 2 \text{cm}^{2} .


The answer is 11.

Victor Paes Plinio by Victor Paes Plinio , 21, Brazil

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2 solutions

Consider my diagram. The area of the gray region is equal to the area of the square minus the area of the yellow region (trapezoid). We have

A = 25 − 1 2 ( 2 + 5 ) ( 4 ) = 11 A=25-\dfrac{1}{2}(2+5)(4)=\boxed{11}

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Marta Reece
Dec 23, 2017

Side of square is 25 = 5 = \sqrt{25}=5= side of rhombus.

Height of rhombus is 20 5 = 4 \frac{20}5=4

The gray shape can be thought of as made of a rectangle on top, sides 5 and 1, area 5, and triangle on the bottom with height 4 and base from point P to the left.

This triangle is a right triangle with hypotenuse 5 and one side 4, so the remaining side, the base, is 3.

Area of the triangle is 1 2 × 3 × 4 = 6 \frac12\times3\times4=6

Total gray area is 5 + 6 = 11 5+6=\boxed{11}

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