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If the sides of a rectangle touch a circle, prove that the rectangle is a square.

Circles

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Answer

The figure below shows a rectangle ABCD with its sides touching the circle at points P, Q, R and S.

If the sides of a rectangle touch a circle, prove that the rectangle is a square. Circles, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

We know that,

Length of tangents from an external point to the circle are equal.

Hence,

AP = AS …..(Eq. 1)

BP = BQ …..(Eq. 2)

CR = CQ …..(Eq. 3)

DR = DS …..(Eq. 4)

Adding the above 4 equations,

⇒ AP + BP + CR + DR = AS + BQ + CQ + DS
⇒ AB + CD = AD + BC

But AB = CD and AD = BC (As opposite sides of a rectangle are equal.)

⇒ AB + AB = BC + BC
⇒ 2AB = 2BC
⇒ AB = BC.

∴ AB = BC = CD = DA.

Hence, proved that ABCD is a square.

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