5 Views

Question : The perimeter of a rhombus is $2p$ units, and the sum of the lengths of the diagonals is $m$ units. The area of the rhombus is:

Option 1: $\frac{m^{2}p}{4}$ sq. units

Option 2: $\frac{mp^{2}}{4}$ sq. units

Option 3: $\frac{m^{2} - p^{2}}{4}$ sq. units

Option 4: $\frac{p^{2} – m^{2}}{4}$ sq. units


Team Careers360 15th Jan, 2024
Answer (1)
Team Careers360 21st Jan, 2024

Correct Answer: $\frac{m^{2} - p^{2}}{4}$ sq. units


Solution : Given that, the perimeter of a rhombus is $2p$ units, and the sum of the lengths of the diagonals is $m$ units.
Let $d_1$ and $d_2$ be the diagonal of the rhombus.
The perimeter of a rhombus $=2\sqrt{d_1^2+d_2^2}$
⇒ $2p=2\sqrt{d_1^2+d_2^2}$
On squaring both sides,
$p^2=d_1^2+d_2^2$ ___(i)
We have,
$d_1^2+d_2^2=m$
Squaring both sides,
$(d_1+d_2)^2=m^2$
⇒ $d_1^2+d_2^2+2d_1d_2=m^2$
From equation (i),
$p^2+2d_1d_2=m^2$
⇒ $d_1d_2=\frac{m^2-p^2}{2}$
The area of the rhombus $=\frac{1}{2}d_1d_2$
$=\frac{1}{2} \times \frac{m^2-p^2}{2}$
$=\frac{m^{2} – p^{2}}{4}$
Hence, the correct answer is $\frac{m^{2} – p^{2}}{4}$ sq. units.

SSC CGL Complete Guide

Candidates can download this ebook to know all about SSC CGL.

Download EBook

Know More About

Related Questions

TOEFL ® Registrations 2024
Apply
Accepted by more than 11,000 universities in over 150 countries worldwide
Manipal Online M.Com Admissions
Apply
Apply for Online M.Com from Manipal University
View All Application Forms

Download the Careers360 App on your Android phone

Regular exam updates, QnA, Predictors, College Applications & E-books now on your Mobile

150M+ Students
30,000+ Colleges
500+ Exams
1500+ E-books