Circles · Class 10 Mathematics: 44 flashcards
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1. How many points does a secant share with a circle?
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Two.
2. How many points does a tangent share with a circle?
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Exactly one, the point of contact.
3. What is a chord?
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The part of a secant that lies inside the circle, joining its two crossing points.
4. State Theorem 1 of this chapter.
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The tangent meets the radius at its point of contact squarely, at 90°.
5. Where exactly is the right angle marked?
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At the point of contact, between the tangent and the radius drawn to it.
6. What is the first step of the proof of Theorem 1?
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Take any other point Q on the tangent and note that it must lie outside the circle, so OQ > OP.
7. Which geometric fact finishes the proof?
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The shortest distance from a point to a line is the perpendicular distance, so OP must be the perpendicular.
8. Give the formula for the length of a tangent from an external point.
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$\sqrt{d^2 - r^2}$, where $d$ is the distance of the point from the centre and $r$ the radius.
9. In triangle OAP with tangent PA, which side is the hypotenuse?
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OP, from the centre to the external point, because the right angle is at A.
10. A point is 13 cm from the centre of a circle of radius 5 cm. Find the tangent length.
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$\sqrt{169 - 25} = 12$ cm.
11. A tangent of length 24 cm is drawn to a circle of radius 7 cm. How far is the point from the centre?
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$\sqrt{49 + 576} = 25$ cm.
12. How many tangents pass through a point inside a circle?
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None.
13. How many tangents pass through a point on a circle?
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Exactly one.
14. How many tangents pass through a point outside a circle?
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Exactly two.
15. Why do the tangents at the ends of a diameter never meet?
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Both are perpendicular to the same line, the diameter, so they are parallel.
16. State the converse of Theorem 1.
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If a line through a point of a circle is perpendicular to the radius at that point, then the line is a tangent.
17. When would you use the converse?
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When a question asks you to PROVE that a given line is a tangent.
18. What does 'tangent length' mean?
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The distance from a named external point to the point of contact — a finite segment, not the whole line.
19. Which point of a tangent is closest to the centre?
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The point of contact, since OP is the perpendicular distance.
20. A circle has radius 8 cm and a tangent of length 15 cm is drawn. Find the distance from the centre.
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$\sqrt{64 + 225} = 17$ cm.
21. Can a straight line meet a circle at three points?
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No — never more than two.
22. What is a tangent, described as a limit?
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The limiting position of a secant as its two points of intersection move together.
23. State Theorem 2 of this chapter.
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The two tangents drawn to a circle from one outside point have equal length.
24. Which congruence criterion proves Theorem 2?
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RHS — right angle, hypotenuse, side.
25. Where do the right angles in that proof come from?
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Theorem 1: the tangent is perpendicular to the radius at the point of contact.
26. Which side is the common hypotenuse in the proof?
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OP, the segment from the centre to the external point, shared by both triangles.
27. Which step turns the congruence into PA = PB?
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CPCT — corresponding parts of congruent triangles are equal.
28. What does OP bisect?
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Both the angle between the two tangents and the angle between the two radii.
29. What kind of quadrilateral is OAPB?
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A kite, with OP as its axis of symmetry.
30. What is OP to the chord of contact AB?
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Its perpendicular bisector.
31. A circle has radius 7 cm and P is 25 cm from the centre. Find both tangent lengths.
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Both are $\sqrt{625 - 49} = 24$ cm.
32. State the circumscribing quadrilateral result.
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$AB + CD = BC + DA$ — the sums of opposite sides are equal.
33. How is that result proved?
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Apply Theorem 2 at each of the four vertices and add the four equalities, then regroup.
34. Why must a circumscribing parallelogram be a rhombus?
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It already has AB = CD and BC = DA; adding AB + CD = BC + DA forces AB = BC, so all four sides are equal.
35. Can a 3 by 5 rectangle circumscribe a circle?
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No — $3 + 3 \ne 5 + 5$, so the opposite-sides condition fails.
36. Give the incircle tangent lengths for a triangle with sides a, b, c.
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$s-a$, $s-b$, $s-c$ from the vertices opposite those sides, where $s$ is the semiperimeter.
37. A triangle has sides 14, 10 and 12. Find the three tangent lengths.
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$s = 18$, so they are $18-14 = 4$, $18-10 = 8$ and $18-12 = 6$.
38. How can you check incircle tangent lengths?
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Each side must be the sum of the two tangent lengths at its ends.
39. What is the relation between the angle at the external point and the angle at the centre?
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They are supplementary: $\angle AOB + \angle APB = 180°$.
40. If two tangents meet at 70°, what is the angle between the radii?
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110°.
41. What is the first step in any tangent-figure question?
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Label the two tangent lengths from each external point with the same letter.
42. Which quadrilaterals can have a circle touching all four sides?
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Only those in which the sums of opposite sides are equal — tangential quadrilaterals.
43. Why can't SAS be used to prove Theorem 2?
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The equal angle is not the included angle between the two known equal sides.
44. How many tangents can be drawn from a point on the circle itself?
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Exactly one, so Theorem 2 has nothing to compare and the tangent length is zero.
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