Multiple choice

In a circle with centre $P$, chord $AB$ is smaller than the sum of the side $PA$ and side $PB$ by $4$cm. M is the midpoint of AB. If perimeter of $\triangle PAB$ is $144$ cm, then find the length of side $PM$.

  1. $12$ cm
  2. $13$ cm
  3. $11$ cm
  4. $10$ cm
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A Correct answer
Explanation

Let PA = PB = r. The perimeter of triangle PAB is PA + PB + AB = 2r + AB = 144. We are given AB = (PA + PB) - 4 = 2r - 4. Substituting this, 2r + (2r - 4) = 144, so 4r = 148, r = 37. In right triangle PAM, PM^2 + AM^2 = PA^2. Since AM = AB/2 = (2r - 4)/2 = r - 2 = 35, PM = sqrt(37^2 - 35^2) = sqrt((37-35)(37+35)) = sqrt(2 * 72) = 12.

AI explanation

Since PA and PB are radii and M is the midpoint of chord AB, the perimeter of triangle PAB is 2*PA + AB = 144. The condition that AB is smaller than the sum of sides PA and PB by 4 cm means 2*PA - AB = 4. Solving these two equations gives PA = 37 cm and AB = 70 cm, meaning AM is 35 cm. Applying the Pythagorean theorem to the right triangle PMA gives PM as the square root of (37^2 - 35^2), which is the square root of 144, or 12 cm.

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