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P is a point lying outside a circle. One line through P cuts the circle at A and then at B; a second line through P cuts it at C and then at D. So A and C are the nearer crossings and B and D the farther ones. The arc BD lying farthest from P measures 118°, and the arc AC nearest to P measures 36°. What angle do the two lines make with each other at P?

Competition practiceGeometry
  • A41°
  • B77°
  • C82°
  • D154°

Step-by-Step Solution

From a point OUTSIDE the circle the angle is half the DIFFERENCE of the two intercepted arcs — far arc minus near arc. So the angle at P = ½(118° − 36°) = ½ × 82° = 41°. Taking half the sum, ½(118° + 36°) = 77°, is the rule for chords crossing INSIDE the circle and is the commonest slip here; 82° is the difference with the halving forgotten; 154° is the plain sum.
Answer: A) 41°

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