SECTION E — Long Answer Type Questions [5 Marks Each]
- 5 Marks Q1. Prove that the lengths of tangents drawn from an external point to a circle are equal. Using this theorem, if $PA$ and $PB$ are tangents from an external point $P$ to a circle with center $O$, and the radius of the circle is $5\text{ cm}$ with $OP = 13\text{ cm}$, calculate the length of chord $AB$ and the area of quadrilateral $PAOB$.
- 5 Marks Q2. Prove that a parallelogram circumscribing a circle is a rhombus. If a quadrilateral $ABCD$ circumscribes a circle such that $AB = 8\text{ cm}$, $BC = 7\text{ cm}$, and $CD = 6\text{ cm}$, find the exact length of side $AD$ and prove that $AB + CD = AD + BC$.
- 5 Marks Q3. Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the center. If two tangents are drawn from an external point $P$ to a circle of radius $r$ such that the angle between them is $60^\circ$, find the distance $OP$ in terms of $r$ and find the perimeter of $\triangle PAB$.
- 5 Marks Q4. A triangle $ABC$ has side lengths $AB = 13\text{ cm}$, $BC = 14\text{ cm}$, and $AC = 15\text{ cm}$. A circle is inscribed in $\triangle ABC$, touching the sides $AB$, $BC$, and $AC$ at points $D$, $E$, and $F$ respectively. Find the lengths of the segments $AD$, $BE$, and $CF$, and calculate the radius of the inscribed circle.
- 5 Marks Q5. $XY$ 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 $\angle AOB = 90^\circ$. Show further that the rectangle formed by the radii and the parallel tangents satisfies the Pythagorean relation for $\triangle AOB$.
- 5 Marks Q6. State and prove the Alternate Segment Theorem: "If a line is tangent to a circle and from the point of contact a chord is drawn, then the angles which this chord makes with the given tangent are equal respectively to the angles formed in the corresponding alternate segments." Use this to find an unknown angle in a cyclic triangle where the tangent-chord angle is $55^\circ$.
- 5 Marks Q7. Prove that the radius $r$ of a circle inscribed in a right-angled triangle with legs $a$ and $b$ and hypotenuse $c$ is given by $r = \frac{a + b - c}{2}$. If the legs are $9\text{ cm}$ and $12\text{ cm}$, calculate the inradius and find the exact lengths into which the hypotenuse is divided by the point of contact.
- 5 Marks Q8. Prove that if two chords of a circle intersect inside the circle, the product of the lengths of their segments are equal ($AP \cdot PB = CP \cdot PD$). Two chords $AB$ and $CD$ intersect at point $P$ inside a circle. If $AP = 4\text{ cm}$, $PB = 6\text{ cm}$, and $CP = 3\text{ cm}$, find the length of $PD$ and the total lengths of chords $AB$ and $CD$ if $P$ is at a specific distance from the center.
- 5 Marks Q9. Two concentric circles are of radii $10\text{ cm}$ and $6\text{ cm}$. Find the length of the chord of the larger circle which touches the smaller circle. Prove that the point of contact bisects this chord, and calculate the area of the annular region bounded by the two circles
- 5 Marks Q10. Prove that if a secant and a tangent of a circle are drawn from an external point, the product of the lengths of the external segment and the entire secant segment equals the square of the tangent segment length ($PA \cdot PB = PT^2$). If a tangent $PT = 12\text{ cm}$ and the external secant segment $PA = 8\text{ cm}$, find the entire secant length $PB$ and the length of the internal chord segment $AB$.
- 5 Marks Q11. Prove that equal chords of a circle are equidistant from the center. If two equal chords $AB$ and $CD$ of a circle intersect inside the circle, prove that the segments of one chord are equal to the corresponding segments of the other chord. Calculate the distance of a $24\text{ cm}$ chord from the center of a circle of radius $13\text{ cm}$
- 5 Marks Q12. If two circles intersect at two points, prove that the line through their centers is the perpendicular bisector of their common chord. Two circles of radii $10\text{ cm}$ and $8\text{ cm}$ intersect, and the length of their common chord is $12\text{ cm}$. Find the distance between their centers.
- 5 Marks Q13. Prove that the perpendicular drawn from the center of a circle to a chord bisects the chord, and conversely, the line drawn through the center to bisect a chord is perpendicular to the chord. Apply this to find the length of a chord situated $5\text{ cm}$ away from the center of a circle whose diameter is $26\text{ cm}$.
- 5 Marks Q14. Prove that the angle in a semicircle is a right angle. Extending this property, prove that if the non-parallel sides of a trapezium are equal, then it is a cyclic trapezium. Calculate the interior and opposite angles of such a cyclic trapezium if one base angle is $70^\circ$.
- 5 Marks Q15. Prove that the tangents drawn at the extremities of any chord of a circle make equal angles with the chord. If a chord $AB$ subtends an angle of $120^\circ$ at the center of a circle of radius $r$, find the angle between the tangents drawn at $A$ and $B$, and calculate the perimeter of the quadrilateral formed by the tangents and radii.
- 5 Marks Q16. Prove mathematically that the locus of the center of a circle touching two intersecting lines is the angle bisector of the lines. Discuss how this property is applied in constructing inscribed circles for complex polygonal land plots.
- 5 Marks Q17. Prove that the angle between two tangents drawn from an external point to a circle is bisected by the line joining the external point to the center. If tangents $PA$ and $PB$ are drawn from $P$ to a circle such that $\angle APB = 60^\circ$ and the radius is $7\text{ cm}$, find the length of $OP$ and the area of the kite-shaped quadrilateral $PAOB$.
- 5 Marks Q18. Prove that the sum of either pair of opposite angles of a cyclic quadrilateral is $180^\circ$, and conversely, if a pair of opposite angles of a quadrilateral is supplementary, the quadrilateral is cyclic. A cyclic quadrilateral $ABCD$ has $\angle A = (2x - 10)^\circ$, $\angle B = (3x + 5)^\circ$, $\angle C = (y + 20)^\circ$, and $\angle D = (y - 10)^\circ$. Find the values of $x$ and $y$ and all four angles.
- 5 Marks Q19. Prove that if an equilateral triangle is circumscribed about a circle of radius $r$, the side length of the equilateral triangle is $2\sqrt{3}r$. If the radius of the inscribed circle is $4\text{ cm}$, calculate the exact perimeter and total area of the circumscribed equilateral triangle
- 5 Marks Q20. Two circles touch each other externally at point $C$. A direct common tangent $AB$ touches the two circles at points $A$ and $B$ respectively. Prove that $\angle ACB = 90^\circ$. If the radii of the two circles are $9\text{ cm}$ and $4\text{ cm}$, find the exact length of the common tangent segment $AB$.
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.
Copyright © 2026 Deepa Maths Academy | Global Examination & Academic Portal. All Rights Reserved.