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AC is a chord of a circle. If X and Y are the two points, on the circumference of the circle, lying on the same side of the chord AC such that ∠AXC=58°, then what is the measure of 2∠AYC?
58°
116°
29°
122°
122°
Angles subtended by the same chord (or arc) at any points on the remaining part of the circle (on the same side) are always equal.
AC is a chord of a circle. X and Y are points on the same side of the arc AC. Given angle AXC = 58°.
∠AXC=∠AYC
Angles subtended by the same chord (or arc) at any points on the remaining part of the circle (on the same side) are always equal.
Students often mistake this for the central angle theorem (where the angle at the center is double the angle at the circumference) and attempt to double the angle incorrectly.
Apply Theorem of Angles in the Same Segment
According to the circle theorem, angles subtended by the same chord (or arc) in the same segment of a circle are equal. Thus, the angle subtended by chord AC at point X is equal to the angle subtended by chord AC at point Y.
∠AXC=∠AYC=58°
Calculate the Required Value
The question asks for the measure of 2∠AYC. Substituting the value of ∠AYC=58° into the expression:
2×58°=116°
D is correct because by the theorem of angles in the same segment, ∠AYC must be equal to ∠AXC (58°), making 2×58°=116°.
Always remember the difference between 'angles in the same segment' (they are equal) and 'angles at the center vs circumference' (the center angle is twice the circumference angle).
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