Question

Write True or False justify your answer :

If the side of a rhombus is $10$ cm and one diagonal is $16$ cm, the area of rhombus is $96{\text{cm}}^2$

Answer 1

Step 1 : Find the diagonal of rhombus


Given : Side of rhombus $=10$ cm
Diagonal of rhombus $=16$ cm

Let $ABCD$ be the rhombus with sides
$AB=BC=CD=DA=10$ cm and $BD=16$ cm

As diagonals of rhombus bisects each other at right angle

$\Rightarrow BO=\frac{BD}{2}=8\text{cm}$

In $△AOB$

$(AB{)}^2=(OA{)}^2+(OB{)}^2$

$(OA{)}^2=(AB{)}^2-(OB{)}^2$

$OA=\sqrt{{10}^2-8^2}$

$=\sqrt{100-64}$

$=\sqrt{36}=6$

$\therefore AC=2\times OA=2\times 6=12$

Thus, the length of the other diagonal is $12$ cm.

Step 2 : Find the area of rhombus

Lengths of the diagonals are $16$ cm and $12$ cm

We know that,
$\text{Area of rhombus}=\frac{1}{2}\times d_1\times d_2$

$=\frac{1}{2}\times 12\times 16$

$=\frac{1}{2}\times 192$

$=96$

Thus, the area of rhombus is $96{\text{cm}}^2$

Hence, the given statement is true.