1.

The two diagonals of a rhombus are of lengths 55 cm and 48 cm. If p is the perpendicular height of the rhombus, then which one of the following is correct?1). 36 cm < p < 37 cm2). 35 cm < p < 36 cm3). 34 cm < p < 35 cm4). 33 cm < p < 34 cm

Answer»

Solution:

Let us take ABCD as the rhombus with obtuse angle at D and B. Let us take E as the point where perpendicular from D MEETS AB and we know that DE is the perpendicular height of the rhombus.

Let the two diagonals of the rhombus meet at the point O.

We have two right angle triangles ∆AOB and ∆DEB. These two triangles are SIMILAR as they have the same angle B and the perpendicular of rhombus bisect each other at right angle.

So,∆AOB ~ ∆DEB (so, their sides are PROPORTIONAL)

AO/DE = AB/DB 

So, DE(perpendicular height) = AO × (DB/AB)

It is given that DB= 48CM and AC=55cm so, AO= 55/2cm

Using PYTHAGORAS theorem in ∆AOB we get,

AB² = AO² + OB²

AB² = (55/2)² + (48/2)²

AB² = 27.5² + 24²

AB² = 756.25 + 576

AB² = 1332.25

AB = √1332.25

AB = 36.5cm

We know DE(p) = AO × (DB/AB)

DE(p) = 27.5 × (48/36.5)

DE(p) = 27.5 × 1.315

DE(p) = 36.15 cm

So,the correct option is 1). 36 cm < p < 37 cm



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