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Why Circles Are Quick Marks
Circle questions look heavy but almost all of them reduce to about six facts. Learn those facts and each question takes under a minute, which is exactly what a timed paper rewards.
The six facts are: the radius meets a tangent at 90°; two tangents from one outside point are equal; a perpendicular from the centre bisects a chord; the angle at the centre is double the angle on the same arc at the circumference; an angle in a semicircle is 90°; and opposite angles of a cyclic quadrilateral add up to 180°.
The usual workflow is simple. Draw the figure, mark every radius, join the centre to the point of contact, and look for a right-angled triangle. Then use Pythagoras. In this short we build each fact with a number you can verify yourself.
1. A point P is 17 cm from the centre of a circle of radius 8 cm. What is the length of the tangent from P to the circle?
2. A chord is 5 cm away from the centre of a circle of radius 13 cm. What is the length of the chord?
3. An arc subtends an angle of 80° at the centre. What angle does it subtend at a point on the remaining part of the circle?
4. ABCD is a cyclic quadrilateral and angle A = 75°. What is angle C?
5. From an external point P, a tangent PT and a secant PAB are drawn, with PA = 4 cm and AB = 5 cm. What is PT?
6. Chords AB and CD of a circle intersect at P inside the circle. If PA = 6 cm, PB = 4 cm and PC = 3 cm, what is PD?
7. A triangle is inscribed in a circle with the diameter as one of its sides. The angle opposite the diameter is:
8. The distance between the centres of two circles of radii 8 cm and 3 cm is 13 cm. What is the length of their direct common tangent?