Mathematics
A quadrilateral ABCD is drawn to circumscribe a circle. Prove that :
AB + CD = AD + BC
Circles
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Answer
We know that,
Tangents drawn from a point to a circle are equal in length.
Therefore,
AS = AP = p (let)
BP = BQ = q (let)
CR = CQ = r (let)
DR = DS = s (let)
To prove :
AB + CD = AD + BC
Solving L.H.S. of above equation :
⇒ AB + CD
⇒ (AP + PB) + (CR + DR)
⇒ p + q + r + s.
Solving L.H.S. of above equation :
⇒ AD + BC
⇒ (AS + DS) + (BQ + CQ)
⇒ (p + s) + (q + r)
⇒ p + q + r + s.
Since, L.H.S. = R.H.S.
Hence, proved that AB + CD = AD + BC.
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