Question

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

Answer 1

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.