Question

If the sides of a quadrilateral ABCD touch a circle, then:

  • AB + BC = CD + AD
  • AB + CD = BC + AD
  • AB - CD = BC - AD
  • AB - CD = BC + AD

✨ Free AI Tools for You

Powered by Chatterbot AI

Answer


The correct option is:

$$ AB + CD = BC + AD $$

Given: A circle touches the sides $AB$, $BC$, $CD$, and $DA$ of quadrilateral $ABCD$ at points $P$, $Q$, $R$, and $S$, respectively.

To Prove: $$ AB + CD = BC + AD $$

Proof:

By Theorem: Tangents drawn from an external point to a circle are equal in lengths.

Since $AP$ and $AS$ are the tangents to the circle from the external point $A$:

$$ AP = AS $$

Similarly:

$$ BP = BQ \ CR = CQ \ DR = DS $$

Adding these equations:

$$ AP + BP + CR + DR = AS + BQ + CQ + DS $$

This simplifies to:

$$ (AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ) $$

Rewriting it, we have:

$$ AB + CD = AD + BC $$

Hence, we conclude:

$$ AB + CD = BC + AD $$


Was this helpful?

India's 1st AI Doubt Solver for CBSE, JEE, and NEET

Ask a Question for Free

and then it's just ₹212 a month