Question

Area of a quadrilateral $ABCD$ is$20c{m}^2$ and perpendiculars on$BD$ from opposite vertices are $1cm$ and $1.5cm$. The length of $BD$ is

• A. $4cm$
• B. $15cm$
• C. $16cm$
• D. $18cm$

Answer 1

The correct option is C

$16cm$

Correct option: (c) $16cm$

Quadrilateral : A quadrilateral is a plane figure that has four sides or edges, and also have four corners or vertices.

Given that, the area of a quadrilateral $=20c{m}^2$

The area of a quadrilateral $ABCD$ can be expressed as the sum of two triangles $△ABD$ and $△CBD.$

Let us assume the altitude $AL$of$△ABD$ is $1.5cm$ and altitude $CM$of$△CBD.$ is $1cm$

The base $BD$is common to both triangles

Then,

$Area(quadABCD)=area△\left(ABD\right)+area△\left(CBD\right)$
$$\begin{gathered} AreaofquadABCD=\frac{1}{2}\times \left(BD\right)\times \left(AL\right)+\frac{1}{2}\times \left(BD\right)\times \left(CM\right) \\ \Rightarrow 20=\frac{1}{2}\times BD\times 1.5+\frac{1}{2}\times BD\times 1 \\ \Rightarrow 20=\frac{1}{2}\times (1.5+1)\times BD \\ \Rightarrow 20=\left(\frac{1}{2}\right)\times (2.5)\times BD \\ \Rightarrow 20\times 2=2.5\times BD \\ \Rightarrow 40=2.5\times BD \\ \Rightarrow BD=\frac{40}{2.5}=16cm \end{gathered}$$

Hence, $BD=16cm$

Hence, the correct option is (c )