Chapter 9.4, Problem 21PSB

To determine

To Calculate: Sum of the lengths of the diagonals.

Answer to Problem 21PSB

The sum of the lengths of the diagonals is $15$

Explanation of Solution

Given information:

The lengths of the diagonals of a rhombus are in the ratio $2:1$

and perimeter of the rhombus is $20$

Formula used:

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

Where

“ $d_1$ ” is the diagonal one of the rhombus

“ $d_2$ ” is the second diagonal of the rhombus

Calculation:

  

Here, $\frac{BC}{AD}=\frac{2}{1}$

And, perimeter of ABCD = $20$

Here, $P=4a$ (“ P ” is perimeter and “ a ” is side)

  →$20=4a$

  →$a=5$

And, area = $a^2$

  →$25$

And, area of rhombus = $\frac{1}{2}\times d_1\times d_2$

  $25=\frac{1}{2}\times d_1\times d_2$

  $25=\frac{1}{2}\times BC\times AD$

  $25=\frac{1}{2}\times 2AD\times AD$

  $AD=5$

So, $BC=10$

And, Sum of diagonal = $AD+BC=5+10$

  →$15$

Conclusion: Hence,the sum of the lengths of the diagonals is $15$