Question

Find the area of the shaded region in figure.

Answer 1

Hint:
First find the area of a right angle triangle $ADB$ using the area of triangle formula $\dfrac{1}{2} \times base \times height$, then find the area of triangle $ABC$, finally by subtracting the area of triangle $ADB$ from the area of triangle $ABC$ we get the area for shaded region.

Complete step by step solution:
From the figure it is shown that the triangle $ABC$ is right angled at point $D$. So we can apply Pythagoras theorem.
Pythagoras theorem is given by:
$Hypotenus{e^2} = Perpendicula{r^2} + Bas{e^2}$
From the figure we can say that:
$AD = $Base
$BD = $Perpendicular or height
$AB = $Hypotenuse
Therefore in the right angle triangle $ADB$
\[A{B^2} = A{D^2} + B{D^2}\]
On substituting the values of base and perpendicular in the above equation, we have
$A{B^2} = {12^2} + {16^2}$
$A{B^2} = 144 + 256$
Taking square root on both the sides
We get, $AB = \sqrt {144 + 256} $
 $ = \sqrt {400} = \sqrt {{{20}^2}} = 20cm$
Therefore $AB = 20cm$
Now, consider triangle $ABC$, by Pythagoras theorem we have,
$A{C^2} = A{B^2} + B{C^2} = {20^2} + {48^2}$
 $ = 400 + 2304$
$ = 2704 = {52^2} = A{C^2}$
Therefore we can say that the triangle $ABC$is right angled at$B$.
Now, to find the area of the shaded region, first we need to find the area of triangle $ADB$ and also the area of triangle $ABC$.
(Area of triangle formula is given by:$\dfrac{1}{2} \times base \times height$)
By using the area of triangle formula we have,
Area of triangle $ADB$ $ = \dfrac{1}{2} \times base \times height$
$ = \dfrac{1}{2} \times 12 \times 16$
$ = 96c{m^2}$
($c{m^2}$Is because when we multiply base in terms of $cm$with height in terms $cm$ we get square of$cm$)
Area of triangle $ABC = \dfrac{1}{2} \times base \times height$
$ = \dfrac{1}{2} \times 20 \times 48$
$ = 480c{m^2}$
Now, to find the area of shaded region as shown in figure
We have,
Area of shaded region $ = $ Area of triangle $ABC$ $ - $ Area of triangle $ADB$
$ = 480 - 96$
$ = 384c{m^2}$

Note:
Pythagoras theorem works only for right-angled triangles. Sometimes they may give or they may not give the value of $AB$ as shown in above fig. If it is given also, if the triangle is a right angled triangle then apply Pythagoras theorem and solve it.