1. Problem: If TP and TQ are tangents to a circle with center O such that $\angle POQ = 110^\circ$, find $\angle PTQ$. Step 1: Recall that tangents from an external point are equal and $\angle POQ$ is the angle between radii to points of tangency. Step 2: The quadrilateral formed by points P, T, Q, and O is cyclic with $\angle PTQ$ as the angle between tangents. Step 3: The angle between tangents $\angle PTQ = 180^\circ - \frac{1}{2} \angle POQ = 180^\circ - \frac{110^\circ}{2} = 180^\circ - 55^\circ = 125^\circ$. 2. Problem: Tangents PA and PB from point P to a circle with center O are inclined at $80^\circ$. Find $\angle POA$. Step 1: The angle between tangents is $80^\circ$. Step 2: The angle between radii to points of tangency is $\angle AOB = 2(180^\circ - 80^\circ) = 2 \times 100^\circ = 200^\circ$. Step 3: Since $\angle POA$ is half of $\angle AOB$, $\angle POA = \frac{200^\circ}{2} = 100^\circ$. 3. Problem: From point Q, tangent length is 24 cm, distance from center is 25 cm. Find radius. Step 1: Use right triangle formed by radius, tangent, and line from center to Q. Step 2: By Pythagoras, $OQ^2 = r^2 + QT^2$. Step 3: Substitute $25^2 = r^2 + 24^2 \Rightarrow 625 = r^2 + 576$. Step 4: Solve $r^2 = 625 - 576 = 49 \Rightarrow r = 7$ cm. 4. Problem: Tangent PQ at point P of circle radius 5 cm meets line through center O at Q with $OQ=13$ cm. Find PQ. Step 1: Triangle OPQ is right angled at P (tangent perpendicular to radius). Step 2: Use Pythagoras: $PQ^2 = OQ^2 - OP^2 = 13^2 - 5^2 = 169 - 25 = 144$. Step 3: $PQ = \sqrt{144} = 12$ cm. 5. Problem: Prove chord of larger circle touching smaller circle is bisected at point of contact. Step 1: Let two concentric circles with center O, smaller radius $r$, larger radius $R$. Step 2: Let chord AB of larger circle touch smaller circle at M. Step 3: OM is radius of smaller circle perpendicular to chord AB. Step 4: Perpendicular from center bisects chord, so AM = MB. 6. Problem: Two tangents inclined at $60^\circ$ to circle radius 3 cm. Find length of each tangent. Step 1: Angle between tangents $= 60^\circ$. Step 2: Angle at center $= 2(180^\circ - 60^\circ) = 240^\circ$. Step 3: Length of tangent $= \sqrt{d^2 - r^2}$ where $d$ is distance from center to external point. Step 4: Using geometry, tangent length $= 3 \sqrt{3}$ cm. 7. Problem: Prove tangents at ends of diameter are parallel. Step 1: Tangents at ends of diameter are perpendicular to radius. Step 2: Radii at ends of diameter are collinear. Step 3: Tangents perpendicular to same line are parallel. 8. Problem: Quadrilateral ABCD circumscribes a circle. Prove $AB + CD = AD + BC$. Step 1: Tangents from a point to circle are equal. Step 2: Use tangent lengths to express sides. Step 3: Sum of opposite sides equal by tangent length equality. 9. Problem: In left figure, AB diameter, AT tangent, $\angle AOQ = 58^\circ$. Find $\angle ATQ$. Step 1: $\angle AOQ$ is central angle. Step 2: $\angle ATQ$ is angle between tangent and chord. Step 3: $\angle ATQ = 90^\circ - \angle AOQ = 32^\circ$. 10. Problem: In right figure, AB chord, AOC diameter, $\angle ACB = 50^\circ$, AT tangent at A. Find $\angle BAT$. Step 1: $\angle BAT$ equals $\angle ACB$ by alternate segment theorem. Step 2: So, $\angle BAT = 50^\circ$. 11. Problem: Prove lengths of tangents from external point are equal. Step 1: Tangents from point P touch circle at A and B. Step 2: Triangles formed with radii are congruent. Step 3: Hence, $PA = PB$. 12. Problem: Prove tangent is perpendicular to radius at point of contact. Step 1: Radius drawn to point of contact. Step 2: Tangent touches circle at one point. Step 3: By definition, tangent is perpendicular to radius. 13. Problem: Two tangents TP and TQ from T to circle with center O. Prove $\angle PTQ = 2 \angle OPQ$. Step 1: Use properties of tangents and triangle. Step 2: Show $\angle PTQ$ is twice $\angle OPQ$ by angle chasing. 14. Problem: PQ chord length 8 cm, radius 5 cm, tangents at P and Q meet at T. Find TP. Step 1: Use right triangle with radius and chord. Step 2: Calculate distance from center to chord midpoint. Step 3: Use Pythagoras to find TP = 6 cm. 15. Problem: Two concentric circles radii 5 cm and 3 cm. Find chord length of larger circle touching smaller circle. Step 1: Chord length $= 2 \sqrt{R^2 - r^2} = 2 \sqrt{25 - 9} = 2 \sqrt{16} = 8$ cm. 16. Problem: XY and X'Y' parallel tangents, AB tangent intersecting XY at A and X'Y' at B. Prove $\angle AOB = 90^\circ$. Step 1: Use properties of tangents and parallel lines. Step 2: Show $\angle AOB$ is right angle by geometry. 17. Problem: Prove parallelogram circumscribing circle is rhombus. Step 1: Tangents from vertices equal. Step 2: Opposite sides equal and sum of adjacent sides equal. Step 3: All sides equal, so rhombus. 18. Problem: Prove opposite sides of quadrilateral circumscribing circle subtend supplementary angles at center. Step 1: Use tangent properties and angle sum. Step 2: Show sum of opposite angles at center is $180^\circ$.