Question

Find the area of a parallelogram given in the figure .Also find the length of the altitude from vertex $ A$on the side $ DC$.

12em 17cem 

Answer 1

Step 1: Writing the given details:

In $ABCD$ parallelogram, $BC=17cm$, $DC=12cm$, $BD=25cm$

$\because$$AB=DC$, therefore, $AB=12cm$

$\because$$AD=BC$, therefore, $AD=17cm$

Step 2: Calculate the perimeter:

Area of $ABCD=$ Area of $∆ADB$ $+$Area of $∆BDC$

For Area $∆BDC$,

$a=DC=12$ , $b=BC=17$,$c=BD=25$

Perimeters $=(DC+BC+BD)÷2$

Perimeter $=\frac{12+17+25}{2}$

Perimeter $=\frac{54}{2}$

Perimeter$=27cm$

Step 3: Calculate the area:

Area of $∆BDC$$=\sqrt{s(s-a)(s-b)(s-c)}$

Area $=\sqrt{27(27-12)(27-17)(27-25)}$

Area $=\sqrt{27\times 15\times 10\times 2}$

Area$=90{cm}^2$

Since,$ABCD$ is a parallelogram,

Area of $∆BDC$$=$ Area of $∆ADB$

Area of parallelogram $ABCD$ $=$ Area of $∆BDC$ $+$ Area of $∆ADB$

Area of parallelogram ABCD $=90+90$

Area of parallelogram$ABCD=180{cm}^2$

Step 3: Find the length of the altitude.

Area of a parallelogram $=$Base $\times$Height

Suppose, the altitude from $A$ be $x$.

Area of parallelogram $ABCD=CD\times$ Altitude from $A$

$\Rightarrow 180=12\times x$

$\Rightarrow x=\frac{180}{12}$

$\Rightarrow x=15cm$

Hence, the area of the parallelogram is$180c{m}^2$and the length of the altitude is $15cm.$