Question

The two diagonals of a rhombus are of length $55cm$ and $48cm$. If $p$ is the perpendicular height of the rhombus, then which one of the following is correct?

• A. $36cm<p<37cm$
• B. $35cm<p<36cm$
• C. $34cm<p<35cm$
• D. $33cm<p<34cm$

Answer 1

The correct option is A $36cm<p<37cm$

Let $ABCD$ be the rhombus with B and D as obtuse angles and perpendicular from D meets AB at E.

Let diagonals $AC$ and $BD$ meet O.
Both $△AOB$ and $\Delta DEB$ are right triangles (as perpendicular of a rhombus bisect each other at right angles and DE is perpendicular on the side AB also angle B is common to both.

Therefore, $△AOB\sim \Delta DEB$
$\Rightarrow \frac{AO}{DE}=\frac{OB}{EB}=\frac{AB}{DB}$ .....CPST
$\therefore DE\left(p\right)=AO\times \frac{DB}{AB}$
As $△AOB$ is right angled, $AB=\sqrt{A{O}^2+B{O}^2}=\frac{73}{2}$
Hence, $p=\left(\frac{55}{2}\right)\times \frac{48}{\frac{73}{2}}=36.164$
$\Rightarrow 36cm<p<37cm$