Question

Find the area of a rhombus whose side is $5cm$ and whose altitude is $4.8cm$. If one of its diagonal is $8cm$ long find the length of the other diagonal.

Answer 1

Step 1: Find the area of the rhombus

All sides of a rhombus are equal. Hence $AB=BC=CD=AD=5cm$

From the figure we can see that,

Area of $\left(ABCD\right)=$Area of $\left(△ACD\right)$$+$Area of $\left(△ABD\right)$

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

We know that the opposite sides of a rhombus are parallel to each other. Hence $AB\parallel CD$

The height for both the triangles is equal to the altitude as the perpendicular distance between parallel lines is constant.

$\Rightarrow$Area of $\left(ABCD\right)=\frac{1}{2}\times CD\times h+\frac{1}{2}\times AB\times h$ where $h=4.8cm$

$\Rightarrow$Area of $\left(ABCD\right)=\frac{1}{2}\times 5\times 4.8+\frac{1}{2}\times 5\times 4.8$

$\Rightarrow$Area of $\left(ABCD\right)=24c{m}^2$

Step 2: Find for the diagonal

The area of a rhombus is also given by the formula

$Area=\frac{1}{2}\times d_1\times d_2$ where $d_1,d_2$are the lengths if the diagonals.

Substituting the values we get

$24=\frac{1}{2}\times 8\times d_2$

$\Rightarrow$$d_2=6cm$

Hence, the area of the given rhombus is $24c{m}^2$ and the length of the other diagonal is $6cm$.