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Q.

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

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a

36 cm < p < 37 cm

b

35 cm < p < 36 cm

c

34 cm < p < 35 cm

d

33 cm < p < 34 cm

answer is A.

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Detailed Solution

We name the given rhombus

ABCD

where

∠ B, ∠ D

are obtuse angles and name the point of intersection of diagonals as O. The perpendicular from D on as DP. We use the fact that diagonals of a rhombus bisect each other perpendicularly and use the similarity of triangles AOB and DPB. We then use Pythagoras theorem to find

p

.
Area of the rhombus =21×55×48 sq. cm
Also area of the rhombus =base×p
∴p=

\frac{55 \times 24}{base}

=

\frac{1320}{base}

base=

\sqrt{{{(27.5)}^{2} + (24) ​}^{2}}

=

\sqrt{1332.25}

= 36.8 cm (lies between 36 and 37)
Therefore, p clearly also lies between 36 and 37.
Hence, option 1 is correct.

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