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CBSE Class 9 Answered

Show that the diagonals of a rhombus are perpendicular to each other.
Asked by Topperlearning User | 04 Jun, 2014, 01:23: PM
answered-by-expert Expert Answer

Given:

ABCD is a rhombus. Diagonal AC and BD intersect at O.

To prove:

AC and BD bisect each other at right angles.

Proof:

A rhombus is a parallelogram such that,

AB = DC = AD = BC ---(i)

Also, the diagonals of a parallelogram bisect each other.

Hence, BO = DO and AO = OC ---(ii)

Now, compare AOB and AOD,

AB = AD (from (i) above)

BO = DO (from (ii) above)

AO = AO (common side)

Therefore, AOB AOD (SSS Congruency Condition)

Therefore, AOB = AOD (corresponding parts)

BD is a straight line segment.

Therefore, AOB + AOD = 1800

But, AOB = AOD (Proved)

AOB = AOD = 1800/ 2 = 900

i.e., the diagonals are perpendicular to each other.

Answered by | 04 Jun, 2014, 03:23: PM
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