Chapter 11.4, Problem 11PSC

To determine

Toprove : The area of any quadrilateral with perpendicular diagonals equals half the product of the diagonals

Explanation of Solution

Given information :

A quadrilateral with diagonals

Formula used :

Area of triangle = $\frac{1}{2}\times \text{base}\times \text{height}$

Proof :

From the figure given, we have

Area of ABCD= Areas of [ triangles AEB+AED+ BEC + BED]

Now,

Area of triangle AEB = $\frac{1}{2}\times yw$

Area of triangle AED = $\frac{1}{2}\times yx$

Area of triangle BEC = $\frac{1}{2}\times zw$

Area of triangle CED = $\frac{1}{2}\times zx$

Area of ABCD = $\left(\frac{1}{2}\times yw\right)+\left(\frac{1}{2}\times yx\right)+\left(\frac{1}{2}\times zw\right)+\left(\frac{1}{2}\times zx\right)$

  $\begin{array}{l}=\frac{1}{2}\left[w\left(y+z\right)+x\left(y+z\right)\right]\\ =\frac{1}{2}\left[\left(w+x\right)\left(y+z\right)\right]\end{array}$

Hence proved