SECTION C — Short Answer Type Questions [3 Marks Each]
- 3 Marks Q26. If the non-parallel sides of a trapezium are equal, prove that it is a cyclic trapezium.
- 3 Marks Q27. Two tangents $TP$ and $TQ$ are drawn to a circle with center $O$ from an external point $T$. Prove that $\angle PTQ = 2 \angle OPQ$.
- 3 Marks Q28. $XY$ 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 $\angle AOB = 90^\circ$.
- 3 Marks Q29. Prove that the parallelogram circumscribing a circle is a rhombus. If the adjacent sides of this parallelogram are in the ratio $3:2$ and perimeter is $50\text{ cm}$, find the side lengths.
- 3 Marks Q30. Prove that the tangents drawn at the extremities of any chord of a circle make equal angles with the chord
- 3 Marks Q31. A circle touches the side $BC$ of $\triangle ABC$ at $P$ and touches $AB$ and $AC$ produced at $Q$ and $R$. If $AB = 5\text{ cm}$, $BC = 6\text{ cm}$, and $CA = 7\text{ cm}$, calculate the lengths of $AQ$, $BP$, and $CP$.
- 3 Marks Q32. Prove that the angle between the two tangents drawn from an external point to a circle is bisected by the line joining the external point to the center.
- 3 Marks Q33. If an equilateral triangle is circumscribed about a circle of radius $r = \sqrt{3}\text{ cm}$, find the side length of the equilateral triangle and its perimeter.
- 3 Marks Q34. Prove that the angle in a semi-circle is a right angle using coordinate or vector geometry principles applied to a circle of radius $r$.
- 3 Marks Q35. If the angle between two radii of a circle is $120^\circ$, find the angle between the tangents drawn at the ends of these radii, and justify using quadrilateral angle sum property.
- 3 Marks Q36. A circle with center $O$ has chords $AB$ and $CD$ of equal length. If $OM \perp AB$ and $ON \perp CD$, prove that $OM = ON$ and explain the congruence condition used.
- 3 Marks Q37. Prove that the locus of the center of a circle touching two intersecting lines is the angle bisector of the lines.
- 3 Marks Q38. If two tangents are drawn to a circle from an external point, prove that the triangle formed by the tangents and the chord of contact is isosceles
- 3 Marks Q39. If a circle is inscribed in a right-angled triangle with sides $9\text{ cm}$, $12\text{ cm}$, and $15\text{ cm}$, find the radius of the inscribed circle and the segments of the hypotenuse created by the point of contact.
- 3 Marks Q40. Two circles touch each other externally at point $C$. A common tangent $AB$ touches the circles at $A$ and $B$. Prove that $\angle ACB = 90^\circ$.
- 3 Marks Q41. Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the center of the circle.
- 3 Marks Q42. A chord $AB$ of a circle of radius $10\text{ cm}$ subtends a right angle at the center. Find the area of the corresponding minor segment and the length of chord $AB$.
- 3 Marks Q43. If an isosceles triangle $ABC$ with $AB = AC = 10\text{ cm}$ is inscribed in a circle of radius $something$ (let radius be $6\text{ cm}$), find the altitude from $A$ to $BC$ and the length of base $BC$.
- 3 Marks Q44. Prove that the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of the circle.
- 3 Marks Q45. In a circle of radius $13\text{ cm}$, a chord is drawn at a distance of $5\text{ cm}$ from the center. Find the length of the chord and the area of the triangle formed by the chord and the radii meeting its endpoints
- 3 Marks Q46. Prove that the circle drawn on any side of a rhombus as diameter passes through the point of intersection of its diagonals.
- 3 Marks Q47. If tangents $PA$ and $PB$ are drawn from an external point $P$ to a circle such that $\angle APB = 60^\circ$ and radius is $5\text{ cm}$, find the length of $PA$ and the area of quadrilateral $PAOB$.
- 3 Marks Q48. Prove that the angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment, and verify it when the angle is $45^\circ$.
- 3 Marks Q49. A triangle $ABC$ has sides $AB = 12\text{ cm}$, $BC = 8\text{ cm}$, and $AC = 10\text{ cm}$. A circle touches $AB$ at $D$, $BC$ at $E$, and $AC$ at $F$. Find the lengths of $AD$, $BE$, and $CF$
- 3 Marks Q50. Prove that the perpendiculars drawn from the vertices of a triangle to the opposite sides (altitudes) form a triangle whose angles are related to the original triangle's angles, and discuss its cyclic properties with the circumcircle.
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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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