Chapter 11.2, Problem 40STP

To determine

To calculate: The lengths of the diagonals of rhombus.

Answer to Problem 40STP

Lengths of the diagonals are 6 mm and 18 mm.

Explanation of Solution

Given information: One diagonal of a rhombus is three times as long as the other diagonal.

Area of rhombus is 54 square millimeters.

Formula used: Area of rhombus using diagonals is given by

  $\text{Area of rhombus = }\frac{1}{2}\times d_1\times d_2$

where, $d_1\&d_2$ are diagonals of rhombus.

Calculation: Since one diagonal of rhombus is three times the other diagonal.

So, Let one diagonal of rhombus be x mm , then other diagonal will be 3 x mm.

Here, $\begin{array}{l}{d}_1=x\text{ mm}\\ d_2=3x\text{ mm}\end{array}$

  $\begin{array}{l}\text{Area of rhombus = }\frac{1}{2}\times d_1\times d_2\\ \Rightarrow 54=\frac{1}{2}\times x\times 3x\\ \Rightarrow 3x^2=108\\ \Rightarrow x^2=36\\ \Rightarrow x=6\text{ mm }\left[\because x=-6\text{ is not possible as }x\text{ is length of the diagonal}\right]\end{array}$

  $\begin{array}{l}\therefore d_1=6\text{ mm}\\ d_2=3\times 6=18\text{ mm}\end{array}$

Thus, lengths of the diagonals are 6 mm and 18 mm.