Multiple choice

Two tangents are drawn to a circle from an external point $A$, touching the circle at the points $P$ and $Q$. A third tangent intersects segment $AP$ at $B$ and segment $AQ$ at $C$ and touches the circle at $R$. If $AQ = 10$ units, then the perimeter of $\Delta ABC$ is

  1. $22.0$
  2. $20.5$
  3. $20.0$
  4. $40.0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

By the property of tangents from an external point, the lengths of tangents from A to the circle are equal (AP = AQ = 10). The perimeter of triangle ABC is AB + BC + AC. Since the third tangent touches at R, BR = BP and CR = CQ. Thus, the perimeter is AB + (BR + RC) + AC = (AB + BP) + (AC + CQ) = AP + AQ = 10 + 10 = 20.

AI explanation

From the external point A, the tangent lengths AP and AQ are equal, making AP equal to 10 units. For the third tangent intersecting at B and C, the equal tangent segments give BR equal to BP and CR equal to CQ. The perimeter of triangle ABC is AB plus BC plus CA, which expands to AB plus BR plus CR plus CA, and by substituting the equal segments, this becomes AB plus BP plus CQ plus CA, ultimately simplifying to AP plus AQ, which is 10 plus 10 to equal 20.0 units.

Keep practicing — related questions