# The following figure shows two concentric squares. Express the area of the shaded triangle as a function of x.

The following figure shows two concentric squares. Express the area of the shaded triangle as a function of x. (#53).

Let, triangle ABC be the shaded triangle.

Observe, the above diagram.

Are of tringle ABC = Are of triangle BCD - Area of tringle ABD.

Area of tringle BCD = 1/2 * BD * CD

From the above diagram, BD = (X - x)/2 and CD = CA + AD = x + [(X - x)/2] = (X + x)/2.

Area of tringle BCD = 1/2 * BD * CD

= 1/2 * (X - x)/2 * (X + x)/2

= (X - x)(X + x)/8

So, area of tringle BCD = (X2 - x2)/8.

Area of tringle ABD = 1/2 * BD * AD

From the above diagram, BD = (X - x)/2 and AD = (X - x)/2

Area of tringle ABD = 1/2 * BD * AD

= 1/2 * (X - x)/2 * (X - x)/2

= (X - x)2/8

So, area of tringle ABD = (X - x)2/8.

Are of tringle ABC = Are of triangle BCD - Area of tringle ABD.

= (X2 - x2)/8 - (X - x)2/8

= [X2 - x2 - X2 - x2 + 2Xx]/8

= [ - 2x2 + 2Xx]/8

= [ - x2 + Xx]/4

= (x/4)[X - x] square units.

Therefore, area of the shaded triangle is (x/4)[X - x] square units.