Exercises
Explore the relationships among chords, arcs, angles, secants, and tangents in circles. This quiz combines theorem recognition with numerical problem-solving, including central and inscribed angles, intersecting chords, tangent-secant lengths, cyclic quadrilaterals, and angles formed inside or outside a circle.
Answer the questions below and check the explanation for each answer.
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The measure of a minor arc equals the measure of the central angle that intercepts it. Therefore, the arc measures 72°.
An inscribed angle measures half its intercepted arc. Half of 110° is 55°.
An angle inscribed in a semicircle is a right angle. Since AB is a diameter, angle ACB measures 90°.
Congruent chords in the same circle intercept congruent arcs. They do not need to be perpendicular or pass through the center.
A radius perpendicular to a chord bisects that chord, dividing it into two congruent segments.
An angle formed by intersecting chords equals half the sum of the intercepted arcs: (80° + 40°) ÷ 2 = 60°.
A radius drawn to a point of tangency is perpendicular to the tangent line, so the angle is 90°.
A tangent-chord angle measures half its intercepted arc. Thus, 140° ÷ 2 = 70°.
An exterior angle formed by two secants equals half the difference of the intercepted arcs: (200° − 80°) ÷ 2 = 60°.
Opposite angles of a cyclic quadrilateral are supplementary. Therefore, the opposite angle is 180° − 112° = 68°.
Among chords in the same circle, the chord closer to the center is longer. Therefore, chord X is longer than chord Y.
For intersecting chords, the segment products are equal: 3 × 8 = 4 × x. Thus, 24 = 4x and x = 6.
The tangent-secant theorem gives t² = 9 × 16 = 144. Taking the positive square root gives t = 12 units.
Inscribed angles in the same segment that subtend the same chord intercept the same arc, so they have equal measures.
The angle between two tangents equals 180° minus the minor arc. Therefore, the minor arc measures 180° − 52° = 128°.
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