SECTION B — Very Short Answer Type Questions [2 Marks Each]
- 2 Marks Q26. Prove that equal chords of a circle subtend equal angles at the center.
- 2 Marks Q27. If chords $AB$ and $CD$ of a circle subtend equal angles at the center, prove that $AB = CD$.
- 2 Marks Q28. The perpendicular from the center of a circle to a chord bisects the chord. Prove this statement for a chord of length $16\text{ cm}$ in a circle of radius $10\text{ cm}$ and find the distance from the center.
- 2 Marks Q29. Prove that the line drawn through the center of a circle to bisect a chord is perpendicular to the chord.
- 2 Marks Q30. Equal chords of a circle are equidistant from the center. If a chord of length $24\text{ cm}$ is at a distance of $5\text{ cm}$ from the center, find the radius of the circle.
- 2 Marks Q31. If two circles intersect at two points, prove that the line through the centers is the perpendicular bisector of the common chord.
- 2 Marks Q32. Prove that the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
- 2 Marks Q33. Angle in a semi-circle is a right angle. Verify this if the diameter has endpoints $( -r, 0 )$ and $( r, 0 )$.
- 2 Marks Q34. Prove that angles in the same segment of a circle are equal
- 2 Marks Q35. If a line segment joining two points subtends equal angles at two other points lying on the same side of the line, the four points are concyclic. State the theorem and its application.
- 2 Marks Q36. Prove that the sum of either pair of opposite angles of a cyclic quadrilateral is $180^\circ$.
- 2 Marks Q37. If the non-parallel sides of a trapezium are equal, prove that it is cyclic.
- 2 Marks Q38. Two chords $AB$ and $CD$ of a circle intersect inside the circle at point $P$. Prove that $AP \cdot PB = CP \cdot PD$.
- 2 Marks Q39. $XY$ and $X'Y'$ are two parallel tangents to a circle with center $O$ and radius $r$. Another tangent $AB$ with point of contact $C$ intersects $XY$ at $A$ and $X'Y'$ at $B$. Prove that $\angle AOB = 90^\circ$.
- 2 Marks Q40. Prove that the angle between the tangent at any point of a circle and the chord through the point is equal to the angle which the chord subtends in the alternate segment
- 2 Marks Q41. If a circle touches the side $BC$ of a triangle $ABC$ at $P$ and touches $AB$ and $AC$ produced at $Q$ and $R$, prove that $AQ = \frac{1}{2}(AB + BC + CA)$.
- 2 Marks Q42. A circle is touching the side $BC$ of $\triangle ABC$ at $P$ and is touched by $AB$ and $AC$ produced at $Q$ and $R$. If $AB = 5\text{ cm}$, $BC = 6\text{ cm}$, and $CA = 7\text{ cm}$, find the lengths of $AQ$, $BP$, and $CP$.
- 2 Marks Q43. Prove that the parallelogram circumscribing a circle is a rhombus. Use this to find the side length if three consecutive sides are proportional.
- 2 Marks Q44. If an equilateral triangle is circumscribed about a circle of radius $r$, find the side length of the equilateral triangle
- 2 Marks Q45. Prove that the lengths of tangents drawn from an external point to a circle are equal using congruent triangles $\triangle OAP$ and $\triangle OBP$.
- 2 Marks Q46. If the angle between two radii of a circle is $130^\circ$, find the angle between the tangents at the ends of these radii.
- 2 Marks Q47. Prove that the tangents drawn at the extremities of any chord make equal angles with the chord.
- 2 Marks Q48. A circle with center $O$ has chords $AB$ and $CD$ of equal length. If $OM \perp AB$ and $ON \perp CD$, prove that $OM = ON$.
- 2 Marks Q49. If a circle is inscribed in a right-angled triangle with legs $a$ and $b$ and hypotenuse $c$, prove that the radius of the inscribed circle is $r = \frac{a + b - c}{2}$.
- 2 Marks Q50. Prove that the locus of the center of a circle touching two intersecting lines is the angle bisector of the lines.
Academic Repository Disclaimer & எச்சரிக்கை அறிவிப்பு :
English: These model solutions are structured to global academic standards for self-assessment and practice purposes. Students are advised to verify with official textbooks for board examinations.
தமிழ்: இந்த மாதிரி கணிதத் தீர்வுகள் மாணவர்களின் சுய கற்றல் மற்றும் பயிற்சித் திறனுக்காக உலகளாவிய கல்வித் தரத்தில் உருவாக்கப்பட்டுள்ளன. தேர்வுகளுக்குத் தயாராகும் போது உங்கள் அதிகாரப்பூர்வ பாடப்புத்தகத்துடன் சரிபார்க்கவும்.
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