Multiple choice

A rectangular display board has corners A(0, 0), B(60, 0), C(60, 20), and D(0, 20), measured in cm. A circular arc passes through points A and C and meets side AB again at P, where AP = 50 cm. What is the radius of the circle?

  1. 25 cm

  2. 25 2 cm

  3. 50 cm

  4. 45 cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Points are A(0,0), B(60,0), C(60,20). A circle passing through (0,0) and (60,20) has its center on the perpendicular bisector. Given AP=50 on the x-axis, P is (50,0). The circle passes through (0,0), (60,20), and (50,0). Solving for the circle equation x^2 + y^2 + Dx + Ey + F = 0 yields the radius.

AI explanation

Since the circle passes through A(0, 0) and P(50, 0), the perpendicular bisector of chord AP gives the x-coordinate of the center as 25. Let the center of the circle be (25, k) and the radius be r, which makes the equation of the circle x squared plus y minus k squared equals r squared. Substituting point A gives 625 plus k squared equals r squared, and substituting point C(60, 20) gives 1225 plus k minus 20 squared equals r squared. Equating both expressions gives 625 plus k squared equals 1225 plus k squared minus 40k plus 400, which simplifies to 40k equals 1000, so k is 25. The radius is the distance from the center (25, 25) to the origin, calculated as the square root of (25 squared plus 25 squared), which is 25 root 2 cm.

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