Mathematics
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.
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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