Question

Prove that the circle drawn with any side of a rhombus as a diameter, passes through the
point of its diagonals.
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Answer 1

Let ABCD be a rhombus having two diagonals AC and BD which bisect each other at right angles.

We have to prove that circle drawn on diameter AB will pass through O.

Now from Q, draw PO || Ad and EF || AB

Again from figure,

AB = CD

=> AB/2 = CD/2

=> AO = DP (since O and P are the mid points of AB and CD)

Similarly AE = OQ

=> AO = OQ = OB

=> A circle drawn with O as center and radius OA passes through A, Q, B.

So the circle drawn with any side of a rhombus as diameter, passes through the point of intersection of its diagonals.

Hence proved.

Answer 2

Answer:

Let ABCD is a rhombus having two diagonals AC and BD which bisect each other at right angles.

We have to prove that circle drawn on diameter AB will pass through O.

Now from Q, draw PO || Ad and EF || AB

Again from figure,

AB = CD

=> AB/2 = CD/2

=> AO = DP (since O and P are the mid points of AB and CD)

Similarly AE = OQ

=> AO = OQ = OB

=> A circle drawn with O as center and radius OA passes through A, Q, B.

So the circle drawn with any side of a rhombus as diameter, passes through the point of intersection of its diagonals.

Hense proved

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