Circle (Myanmar Exam Board)

 

Group (2015-2019)

1. (2015/Myanmar /q12a )
Two circles cut at $A, B$. The tangent to the first at $A$ meets the second again at $C$; and the tangent to the second at $B$ meets the first again at $D$. Prove that $A D$ and $C B$ are parallel. $\quad$. (5. marks)

2. (2015/Myanmar /q12b )
$A B C$ is a triangle in which $A B=A C$. $P$ is a point insidè the triangle such that $\angle P A B=\angle P B C \cdot Q$ is the point on $B P$ such that $A Q=A P$. Prove that $A B C Q$ is cyclic. (5 marks )

3. (2015/FC /q12a )
In the figure, $A B C D E$ is a semicircle at centre $O$, the segment $A E$ is the diameter and $B, C, D$ are any points on the arc. Prove that $\angle A B C+\angle C D E=270^{\circ}$.

4. (2015/FC /q12b )
Given: $\angle A B E=\angle A D E$ and $\angle D A C=\angle D E C$. Prove: $A, B, C, D$ and $E$ all lie on one circle.(5 marks)

5. (2016/Myanmar /q4 )
Given : $\odot O$ with $A B=A D$ and $A C$ is a diameter. $$\text { Prove : } B C=C D$$

6. (2016/Myanmar /q12a )
Prove that the opposite angles of a quadrilateral inscribed in a circle are supplementary.

7. (2016/Myanmar /q12b )
$A B C D$ is a parallelogram. Any circle through $A$ and $B$ cuts $D A$ and $C B$ produced at $P$ and $Q$ respectively. Prove that $D C Q P$ is cyclic.

8. (2016/FC /q4 )
$A$ and $B$ are two points on a circle $3 \mathrm{~cm}$ apart. The chord $A B$ is produced to $C$ making $B C=1 \mathrm{~cm}$. Find the length of the tangent from $C$ to the circle.

9. (2016/FC /q12a )
In the figure, $Q P T$ is a tangent at $P$ and $P D$ is a diameter. If $\angle B P T=x, \operatorname{arc} D C=\operatorname{arc} C B$ then find $\angle D P C, \angle C P B$ and $\angle Q P C$ in terms of $x$.

10. (2016/FC /q12b )
Two incongruent circles $P$ and $Q$ intersect at $A$ and $D$, a line $B D C$ is drawn to cut the circle $P$ at $B$ and circle $Q$ at $C$, and such that $\angle B A C=90^{\circ}$. Prove that $A P D Q$ is cyclic.

11. (2017/Myanmar /q4 )
In the figure, $O$ is the centre of the circle, find $\angle R Q T$.

12. (2017/Myanmar /q12a )
Two unequal circles are tangent internally at $A ; B C$, a chord of the larger circle, is tangent to the smaller circle at $D ;$ prove that $A D$ bisects $\angle B A C$.

13. (2017/FC /q4 )
In circle $\mathrm{O}, \mathrm{PS}$ is a diameter and $\angle \mathrm{POQ}=60^{\circ}, \angle \mathrm{ROS}=70^{\circ}$. find $\angle \mathrm{PTQ} .$

14. (2017/FC /q12a )
In the figure, $O$ is the centre of the circle, $\mathrm{AFG} / / \mathrm{OB}, \angle \mathrm{AOB}=120^{\circ}$ and $\angle \mathrm{EAG}=80^{\circ}$. CC Find $\angle \mathrm{BFG}$ and $\angle \mathrm{EBO}$. $\mathrm{A} \longrightarrow \mathrm{F}_{\mathrm{G}} \quad(5 \mathrm{marks})$

15. (2017/FC /q12b )
In $\triangle \mathrm{ABC}, \mathrm{AB}=\mathrm{AC} . \mathrm{P}$ is any point on $\mathrm{BC}$, and $\mathrm{Y}$ any point on $\mathrm{AP}$. The circle BPY and CPY cut $\mathrm{AB}$ and $\mathrm{AC}$ respectively at $\mathrm{X}$ and $\mathrm{Z}$. Prove $\mathrm{XZ} / / \mathrm{BC} . \quad$ ( $5 \mathrm{marks})$

16. (2018/Myanmar /q4 )
In the figure, $\angle B C D=125^{\circ}$ find $x$ and $y$.

17. (2018/Myanmar /q12a )
Through the points of intersection of two circles, two straight lines $A B$ and $C D$ are drawn meeting one circle at $A, C$ and the other at $B, D$. Prove that $A C / / B D$.

18. (2018/Myanmar /q12b )
$A B C$ is a triangle inscribed in a circle and $D E$ the tangent at $A$. A line drawn parallel to $D E$ meets $A B, A C$ at $F, G$ respectively. Prove that $B F G C$ is a cyclic quadrilateral.

19. (2018/FC /q4 )
Given : $A B C D E F$ is an inscribed regular hexagon. $P F$ is a tangent to the circle $O$ at $F$. Prove : $P F$ and $E A$ are parallel.

20. (2018/FC /q12a )
If $A, B, C$ are three points on the circumference of a circle such that the chord $A B$ is equal to the chord $A C$, prove that the tangent at $A$ bisects the exterior angle between $A B$ and $A C$.

21. (2018/FC /q12b )
Two circles intersect at $A$ and $B$. $A$ point $P$ is taken on one so that $P A$ and $P B$ cut the other at $Q$ and $P$ respectively. The tangents at $Q$ and $R$ meet the tangent at $P$ in $S$ and $T$ respectively. Prove that $\angle T P R=\angle B R Q$ and $P B Q$. is cyclic.

22. (2019/Myanmar /q4a )
AT and BT are tangents to the circle $A B C$ at $\mathrm{A}$ and $\mathrm{B}$. Prove that $\angle \mathrm{BTX}=2 \angle \mathrm{ACB}$. (3 marks)

23. (2019/Myanmar /q11a )
PT is a tangent and PQR is a secant to a circle. A circle with $\mathrm{T}$ as centre and radius TQ. meets QR again at S. Prove that $\angle \mathrm{RTS}=\angle \mathrm{RPT} . \quad$ (5 marks)

24. (2019/Myanmar /q12a )
Prove that the quadrilateral formed by producing the bisectors of the interior angles of any quadrilateral is cyclic. : $\quad$ ( 5 marks)

25. (2019/FC /q4a )
Given: $\mathrm{XY}$ is the tangent at C. Prove: $\mathrm{XY} / / \mathrm{DE}$

26. (2019/FC /q11a )
$\mathrm{ABC}$ is an acute-angled triangle inscribed in a circle whose centre is $\mathrm{O}$, and $\mathrm{OD}$ is the perpendicular drawn from $\mathrm{O}$ to $\mathrm{BC}$. Prove $\angle \mathrm{BOD}=\angle \mathrm{BAC}$. $(5 \mathrm{marks})$

27. (2019/FC /q12a )
Two circles cut at $\mathrm{C}, \mathrm{D}$ and through $\mathrm{C}$ any line $\mathrm{ACB}$ is drawn to meet the circles at $\mathrm{A}$, B. AD and BD are joined and produced to meet the circles again at E, F. If AF, BE produced meet at $\mathrm{G}$, prove that $\mathrm{D}, \mathrm{F}, \mathrm{G}, \mathrm{E}$ are concyclic. (5 marks)

28. (2017/Myanmar /q12b )
$P V$ is a tangent to the circle and $Q T$ is parallel to $P V$ Prove that $Q R S T$ is a cyclic quadrilateral.

Answer
1.  Prove 
2.  Prove
3.  Prove 
4.  Prove
5.  Prove
6.  Prove 
7.  prove  
8.  $2 \mathrm{~cm}$
9.  $\angle D P C=\frac{90-x}{2}=\angle C P B, \angle Q P C=\frac{270-x}{2}$
10.  Prove
11.  $\angle Q R T=62$
12.  Prove 
13.  $65^{\circ}$
14.  $60^{\circ},40^{\circ}$ 
15.  Prove
16.  $x=35, y=55$
17.  Prove 
18.  Prove
19.  Prove
20.  Prove 
21.  Prove
22.  Prove
23.  Prove
24.  Prove
25.  Prove 
26.  Prove 
27.  Prove 
28.  Prove

Group (2014)

1. PTU is the tangent at the point $T$ to the circle and $P Q R$ is a straight line. If $\angle P Q T=100^{\circ}, \angle U T S=55^{\circ}$ and $P Q=Q T$, find $\angle T S R, \angle T R Q$ and $\angle S T R .$ (5 marks)


2. In the figure, $O$ is the centre of the circle and $\angle A O Q=90^{\circ}$. Prove that $\angle O P A=\angle O Q B$. (5 marks)   


3. $A C$ and $B D$ are chords of a circle. Given that $\angle B P C=72^{\circ}, \angle P C D=18^{\prime}$ and $C P=C B$, find $\angle P D C, \angle A B P$ and show that $A C$ is a diameter. (5 marks)


4. In the figure, $O$ is the centre of the circle. $A C=C B$ and $A E$ is the tangent at $A$ which meets $B D$ produced at $E$. Given that $\angle E A D=32^{\circ}$, calculate $\angle B O C$ and $\angle A E D .$  (5 marks)


5. OA and $O B$ are two radii in $\odot O$. In the figure, if $A C / / B D$ and $\angle A O B=120^{\circ}$, prove that $\triangle A E C$ is equilateral. (5 marks)


6. $A B C D$ is a quadrilateral inscribed in a circle whose centre is $O . A B$ is a diameter of the circle. If $B C=C D$, prove that $\angle B D C=\angle C A D$, $\angle B O D=4 \angle C A D$ and $\angle A B D+2 \angle D B C=90^{\circ}$. (5 marks)

7. Two circles intersect at $A$ and $B$. At A a tangent is drawn to each circle meeting the circles again at $P$ and $Q$ respectively. Prove that $\angle A B P=\angle A B Q$ and $A B^{2}=B P \cdot B Q$. (5 marks)

8. Two circles intersect at $P$ and $Q .$ At $P$, a tangent is drawn to each circle meeting the circles again at $X$ and $Y$ respectively. Prove that $\angle P Q X=\angle P Q Y$ and $P Q^{2}=Q X \cdot Q Y$. (5 marks)

9. $A T$ is a tangent at $T$ and $A B C$ is a secant to a circle. A circle with $T$ as center and radius $T B$ meets $B C$ again at $D .$ Prove that $\angle C T D=\angle C A T$. (5 marks)

10. A triangle is inscribed in a circle. Prove that the sum of the measures of the angles in the segments exterior to the triangle is four right angles. (5 marks)

11. A circle passes through the vertex $A$ of an equilateral triangle $A B C$ and is tangent to $B C$ at its midpoint $D$. Find $A E: E C .$ (5 marks)


12. In the figure, $F G$ is a diameter and $H I$ is the tangent at $G$. Prove that $L, H, I, M$ are concyclic. (5 marks)


13. In the figure, $A B$ is a diameter and $C D$ is the tangent at $B$. Prove that $\frac{A C}{A D}=\frac{A F}{A E}$. (5 marks)


14. $A B$ is a diameter of a circle and $C D$ is a tangent at $B \cdot A C$ and $A D$ cut the circle at $G$ and $H$ respectively. Prove that $A C \cdot A G=A D \cdot A H$. (5 marks)

15. In $\triangle A B C, A X, B Y, C Z$ are the perpendiculars from the vertices to the opposite sides. If the perpendiculars meets at $O$, prove that $A O \cdot O X=B O \cdot O Y=C O \cdot O Z$. (5 marks)

16. $A B C$ is a triangle inscribed in a circle. The tangent at A meets $C B$ produced at $D$ and $A E$ bisecting $\angle C A B$ meets side $C B$ at $E$. Prove that $\triangle D A E$ is isosceles. (5 marks)

17. Through the points of intersection of two circles two straight lines $A B$ and $C D$ are drawn meeting one circle at $A, C$ and the other at $B, D$. Prove that $A C$ parallel to $B D$. (5 marks)

18. Two chords $A B$ and $C D$ of a circle intersect at right-angles at $K, E$ is the midpoint of $K D$. If $A K=6 \mathrm{~cm}, C K=3 \mathrm{~cm}$ and $K D=4 \mathrm{~cm}$, find the length of $B E$. If $A E$ is produced to meet the circle again at $F$, show that $A E=4 E F .$ (5 marks)

19. $O$ is a point inside $\triangle A B C, B O, C O$ produced meet $A C, A B$ and $X, Y$ respectively. If $A X O Y$ and $B Y X C$ are cyclic quadrilateral, prove that $B X$ and $C Y$ are altitude of $\triangle A B C .$ (5 marks)

Answer
1. $\angle T S R=100^{\circ}, \angle T R Q=40^{\circ}, \angle S T R=25^{\circ} \quad$ 
2. Prove
3. $\angle A B P=18^{\circ}, \angle P D C=54$
4. $\angle B O C=58^{\circ}, \angle A E D=26^{\circ}$
5. Prove
6. Prove
7. Prove
8. Prove
9. Prove
10. Prove
11. $3: 1$
12. Prove
13. Prove
14. Prove
15. Prove
16. Prove
17. Prove
18. Prove
19. Prove


Group (2013)

1. Given that $M N O P Q R$ is a hexagon inscribed in a circle, show that $\angle R M N+\angle N O P+\angle P Q R=360^{\circ}$. (5 marks)

2. $A, B$ and $C$ are three points on the circumference of a circle such that $A B=A C$. Prove that the tangent at $A$ bisects the exterior angle between $A B$ and $A C$. (5 marks)

3. If $P, Q, R$ are three points on the circumference of a circle such that the chords $P Q$ is equal to the chord $P R$, prove that the tangent at $P$ bisects the exterior angle between $P Q$ and $P R$. (5 marks)

4. Two unequal circles are tangent externally at $O \cdot A B$ is a chord of the first circle. $A B$ is tangent to the second circle at $C$, and $A O$ meets this circle at $E$. Prove that $\angle B O C=\angle C O E$. (5 marks)

5. $P Q R$ is an acute triangle, a circle is described on the side $Q R$ as a diameter and cuts $P Q$ and $P R$ in $X$ and $Y$ respectively. $Q Y$ and $R X$ intersect in $Z$. Prove that $\angle Q Z R=\angle P Q R+\angle P R Q$. (5 marks)

6. Two unequal circles are tangent internally at $A ; B C$, a chord of the larger circle, is tangent to the smaller circle at $D$; prove that $A D$ bisects $\angle B A C$. (5 marks)

7. Two circles $P$ and $Q$ intersect at $A$ and $D$, two parallel lines $B A C$ and $E D F$ are drawn to cut the circle $P$ at $B, E$ and circle $Q$ at $C, F$ respectively. Show that $B C F E$ is a parallelogram. (5 marks)

8. $P Q R$ is an acute triangle inscribed in a circle whose centre is $O$, and $O S$ is the perpendicular drawn from $O$ to $Q R$. Prove that $\angle Q O S=\angle Q P R$. (5 marks)

9. (Fig)Given $\odot O$ with diameter $C I$, $C M \| O N$, prove that arc $M N=\operatorname{arc} N I$. (5 marks)


10. Two incongruent circles $P$ and $Q$ intersect at $A$ and $D$, a line $B D C$ is drawn to cut the circle $P$ at $B$ and circle $Q$ at $C$, and such that $\angle B A C=90^{\circ}$. Prove that A, $P, D, Q$ are concyclic. (5 marks)

11. Two incongruent circles $P$ and $Q$ intersect at $A$ and $D$, a line $B D C$ is drawn to cut the circle $P$ at $B$ and circle $Q$ at $C$, and such that $\angle B A C=90^{\circ}$. Prove that $A P D Q$ is cyclic. (5 marks)

12. Two unequal circles intersect at $P$ and $Q$ with their centres on opposite sides of the common chord $P Q$. Through $P$ the diameters $P A$ and $P B$ are drawn. The tangents at $A$ and $B$ meet at $C$. Prove that $A Q B$ is a straight line. Prove also that a circle can be drawn through the points $A, P, B$ and $C$. (5 marks)

13. Two circles intersect at $A$ and $B$. A point $P$ is taken on one so that $P A$ and $P B$ cut the other at $Q$ and $R$ respectively. The tangents at $Q$ and $R$ meet the tangent at $P$ in $S$ and $T$ respectively. Prove that $\angle T P R=\angle B R Q$ and $P B Q S$ is cyclic. (5 marks)

14. $P Q R S$ is a parallelogram. A circle through $P, Q$ cuts the diagonals $P R, Q S$ at $A$, $B$ respectively. Prove that $A, B, S, R$ are concyclic. (5 marks)

15. $A B C D$ is a parallelogram. Any circle through $A$ and $B$ cuts $D A$ at $P$ and $C B$ produced at $Q$. Prove that $D C Q P$ is cyclic. (5 marks)

16. In $\triangle A B C, L, M, N$ are the midpoints of the sides $A B, B C, A C$ respectively. If $A D \perp B C$, show that $D, L, M, N$ are concyclic. (5 marks)

17. (Fig)In the figure, $X$ is the mid-point of the chord $A B$ and $X Y$ is parallel to $A T$, the tangent at $A$. Prove that $\angle A Y X=\angle A B C$ and $B X Y C$ is a cyclic quadrilateral. Prove also that $A B^{2}=2 A Y A C$. (5 marks) 


18. (Fig)Circles $P$ and $Q$ are congruent and tangent externally at $O$. Prove that $O A=O B$. (5 marks)


19. $A B C$ is √° triangle in which $A B=A C . P$ is a point inside the triangle such that $\angle P A B=\angle P B C$. Given that $Q$ is a point on $B P$ produced such that $A B C Q$ is a cyclic quadrilateral, prove that $A Q=A P$. (5 marks)

20. $A B$ is a diameter of a circle and $E$ any point on the circumference. From any point $C$ on $A B$ produced, a line is drawn perpendicular $A B$, meeting $A E$ produced at $D$. Prove that $A E \cdot A D=A B \cdot A C$. (5 marks)

21. A circle passes through the vertex $A$ of an equilateral triangle $A B C$ and is tangent to $B C$ at its midpoint $D($, cuts $A C$ at $E)$. Find $A E: E C$. (5 marks)

Answer (2013) 

21. 3:1

Group (2012)



$\quad$ $\,$
1.$A B C$ is a triangle inscribed in a circle. The tangent at $A$ meets $C B$ produced at $T$ and $P$ is a point on $B C$ such that $T A=T P$. Prove that $\angle B A P=\angle C A P$. (5 marks)
2.$P Q R$ is a triangle inscribed in a circle. The tangent at $P$ meets $R Q$ produced at $T$, and $P C$ bisecting $\angle R P Q$ meets side $R Q$ at $C$. Prove $\Delta T P C$ isosceles. (5 marks)
3.In $\triangle A B C, A B=A C . P$ is any point on $B C$, and $Y$ is any point on $A P$. The circles $B P Y$ and $C P Y$ cut $A B$ and $A C$ respectively at $X$ and $Z$. Prove $X Z / / B C$. (5 marks)
4.$P Q R S$ is a cyclic quadrilateral, $Q R$ and $P S$ are produced to meet at $E$. If $\angle E S R=\angle R P Q$, then.prove that $Q R=P R$ and $\angle Q S R=\angle R S E .$ (5 marks)
5.$P Q R$ is a triangle inscribed in a circle and $A B$ is the tangent at $P .$ A line $C D$ is drawn to meet $P Q, P R$ at $C, D$ respectively. If $C D R Q$ is cyclic, show that $A B / / C D$. (5 marks)
6.$A B C$ is a triangle in which $A B=A C . P$ is a point inside the triangle such that $\angle P A B=\angle P B C . Q$ is the point on $B P$ such that $A Q=A P$. Prove that $A B C Q$ is cyclic. (5 marks)
7.In $\triangle P Q R, X, Y, Z$ are the middle points of $P Q, P R, Q R$ respectively, and $O$ is the point of perpendicular from $Q$ to $P R$. Then prove that $X, Z, Y, O$ are concylic. (5 marks)
8.In a parallelogram $A B C D, A M \perp B C$ and $C N \perp A B$. If $A M$ and $C N$ intersect at $E$, show that $A, C, D$ and $E$ are concyclic. (5 marks)
9.If $L, M, N$ be the middle points of the sides of a $\triangle A B C$, and it $P, Q, R$ be the feet of the perpendiculars from the vertices on the opposite sides, prove $P, N, Q, L, M, R$, are concyclic. (5 marks)
10.$A B C D$ is a square and $E$ the middle point of $C D$. A circle drown through $A, B$ and $E$ meets $B C$ at $F$. Prove $C F=\frac{1}{4} C B$. (5 marks)
11.$A B C D$ is a square and $E$ the middle point of $C D$. A circle drawn through $A, B$ and $E$ meets $B C$ at $F$. Prove that $B C=4 C F$. (5 marks)
12.The line $T C B$ cuts a circle at $C$ and $B$ and the line $T A$ is a tangent to the circle at $A$. Given that $A B=A T$ prove that $C A=C T$. Given also that $B C$ is a diameter of the circle, calculate $\angle A T C$. (5 marks)
13.$P, Q, R, S$ are points on a circle. $P R$ and $Q S$ cross at $X$, if $P X=10 \mathrm{~cm}, S X=4 \mathrm{~cm}$, $R S=2.5 \mathrm{~cm}$, then find $P Q$. (5 marks)
14.$P Q$ is a chord of a circle and $R$ is any point on the major arc $P Q$. Two chords $P Q$. and $R S$ intersect at $A$. The circle tangent to $P Q$ at $P$ and passes through $R$ cuts $R S$ at $B$. If $A B=A S$, then show that $P A=A Q$. (5 marks)
15.Through the points of intersection of two circles two straight lines $A B$ and $C D$ are drawn meeting one circle at $A, C$ and the other at $B, D$. Prove that $A C \| / B D$. (5 marks)
16.Iwo circle cut at $A, B$. The tangent to the first at $A$ meets the second again at $C$; and the tangent to the second at $B$ meets the first again at $D .$ Prove that $A D$ and $C B$ are parallel. (5 marks)
17.In the figure $Q P T$ is a tangent at $P$ and $P D$ is a diameter. If $\angle B P T=x, \operatorname{arc} D C=\operatorname{arc} C B$ then find $\angle D P C$ and $\angle C P B$ and $\angle Q P C$ in terms of $x$. (5 marks)
18.The tangent at the point $C$ on a circle meets the diameter $A B$ produced at $T$. If $\angle B C T=27^{\circ}$, calculate $\angle C T A$. If $C T=t$ and $B T=x$, prove that the radius of the circle is $\frac{t^{2}-x^{2}}{2 x}$. (5 marks)


Answer (2012)



$\quad$ $\,$
1.Prove
2.Prove
3.Prove
4.Prove
5.Prove
6.Prove
7.Prove
8.Prove
9.Prove
10.Prove
11.Prove
12.$30^{\circ}$
13.$6 .25 \mathrm{~cm}$
14.Prove
15.Prove
16.Prove
17.$\frac{90^{\circ}-x}{2}, \frac{90^{\circ}-x}{2}, \frac{270^{\circ}-x}{2}$
18.$36^{\circ}$


Group (2011)


$\quad\;\,$$\,$
1.In figure if $A P=10, P D=6$, $D A=12, B C=9$, find $A B$ and $C D$. $\mbox{ (5 marks)}$


2.In the figure $A C$ is tangent to the $\odot A B D$; $C B D$ and $D A E$ are straight lines. Find $B D$ and $A E$ using the given data in the figure. $\mbox{ (5 marks)}$


3.$A B C$ is a triangle inscribed in a circle whose centre is $O$, and $O D$ is the perpendicular drawn from $O$ to $B C$.Prove that $\angle B O D=\angle B A C$. $\mbox{ (5 marks)}$

4.Two circles intersect at $A, B$.At A a tangent is drawn to each circle meeting the circles again at $P$ and $Q$ respectively.Prove that $\angle A B P=\angle A B Q$. $\mbox{ (5 marks)}$

5.Two circles intersect at $P ; Q \cdot$ At $P$, a tangent is drawn to each circle meeting the circles again at $R$ and $S$ respectively.Prove that $\angle P Q R=\angle P Q S$. $\mbox{ (5 marks)}$

6.$O A$ and $O B$ are two radii of a circle meeting at right angles.From $A, B$ two parallel chords $A X, B Y$ are drawn.Prove that $A Y \perp B X$. $\mbox{ (5 marks)}$

7.In the figure $A T$ is a tangent at $A$ and $A C$ is a chord.If $B$ is the middle point of arc $A B C$, prove that $A B$ bisects $\angle C A T$, and the perpendiculars from $B$ to the tangent and the chord are equal. $\mbox{ (5 marks)}$


8.Two circles intersect at $A$ and $B$.At $A$ a tangent is drawn to each circle meetin\xi the circles again at $P$ and $Q$ respectively. Prove that $\angle A B P=\angle A B Q$.Prove alsc that $A B^{2}=B P \cdot B Q$. $\mbox{ (5 marks)}$

9.$A B C$ is a triangle inscribed in a circle and $D E$ the tangent at $A$.A line drawn parallel to $D E$ meets $A B, A C$ at $F, G$ respectively.Prove that $B F G C$ is a cyclic quadrilateral. $\mbox{ (5 marks)}$

10.Two circles cut at $A$ and $B$.Through $A$ a line $C A D$ is drawn to meet the circles at $C$ and $D. C B$ and $D B$ are joined and produced to meet the circles again at $E$ and $F$. If $C F$ produced and $D E$ produced meet at $G$, prove that the points $B, F, G, E$ are concyclic. $\mbox{ (5 marks)}$

11.$A B C D$ is a parallelogram.A circle through $A, B$ cuts $B C, A C, B D$ and $A D$ at $H, Q, P, K.$ Prove that $C, D, H, K$ are concyclic. $\mbox{ (5 marks)}$

12.Given: $\angle A B E=\angle A D E$ and $\angle D A C=\angle D E C$ Prove: $A, B, C, D$ and $E \mathrm{a}^{1}$ llie on one circle. $\mbox{ (5 marks)}$


13.From a point $D$ on the base $B C$ of $\triangle A B C$ a line is drawn meeting $A B$ at $E$ and such that $\angle B D E=\angle A.$ Prove that $B E \cdot B A=B D \cdot B C$. $\mbox{ (5 marks)}$

14.$\triangle P Q R$ is a triangle in which $P L, Q M$ and $R N$ are the perpendiculars drawn from the vertices to the opposite sides.If the perpendiculars meet at $O$, prove that $P O \cdot O L=Q O \cdot O M=R O \cdot O N$. $\mbox{ (5 marks)}$

15.In the figure, $A B$ is a diameter and $C D$ is the tangent at $B$.Prove that $\frac{A C}{A D}=\frac{A F}{A E}$. $\mbox{ (5 marks)}$




Answer (2011)


$\quad\;\,$$\,$
1.$\mathrm{AB}=14.5, \mathrm{CD}=13.5$
2.$\mathrm{BD}=2.2, \mathrm{AE}=4.2$
3.Prove
4.Prove
5.Prove
6.Prove
7.Prove
8.Prove
9.Prove
10.Prove
11.Prove
12.Prove
13.Prove
14.Prove
15.Prove


Group (2010)


$\quad\;\,$$\,$
1.(figure)Given $\odot O$ with $A C=B D$, prove that $\triangle P B C$ is isosceles with base $B C.$ $\text{ (5 marks)}$


2.In a circle with centre $S$, $AB$ and $BC$ are congruent chords. $SV$ and $SU$ are two radii such that $S V \perp A B$ and $S U \perp B C.$ Prove that $B$ is the midpoint of arc $V U.$ $\text{ (5 marks)}$

3.(figure)Circles $P$ and $Q$ are congruent and tangent externally at $O.$ Prove that $O A=O B.$ $\text{ (5 marks)}$


4.Two circles intersect at $M, N$ and from $M$ diameters $M A, M B$ are drawn in each circle.If $A, B$ be joined to $N$, prove that $A N B$ is a straight line. $\text{ (5 marks)}$

5.(figure)Given $\odot O$ with diameter $C I$, $C M \| O N$, prove that $\operatorname{arc} M N=\operatorname{arc} N I.$ $\text{ (5 marks)}$


6.(figure)Given $\odot G$ with diameter $C I$, $C M \| G N$, prove that $\operatorname{arc} M N=\operatorname{arc} N I.$ $\text{ (5 marks)}$


7.Draw a circle and a tangent, $T A S$ meeting it at $A.$ Draw a chord $A B$ making $\angle T A B=60^{\circ}$ and another chord $B C \| T S.$ Prove that $\triangle A B C$ is equilateral. $\text{ (5 marks)}$

8.Draw a circle and a tangent $T A S$ meeting it at $A.$ Draw a chord $A B$ making $\angle T A B=30^{\circ}$ and another chord $B C \| T S.$ Prove that $A B=A C$ and find $\angle B A C.$ $\text{ (5 marks)}$

9.$P Q R S$ is a square.$A$ circle through $P, Q$ touches $R S$ at $M$ and cuts $Q R$ at $N$, such that $N R=\frac{1}{4} Q R.$ Prove that $M$ is the middle point of $R S.$ $\text{ (5 marks)}$

10.$A B C$ is a triangle inscribed in a circle.The tangent at $A$ meets $C B$ produced at $D$, and $A E$ bisecting $\angle C A B$ meets side $C B$ at $E.$ Prove that $\triangle D A E$ is isosceles. $\text{ (5 marks)}$

11.$T A$ and $T B$ are tangents to the circle with centre at $O$ and $T A$ produced meets $B O$ produced at $X$ and $X B=4 \mathrm{~cm}, T B=3 \mathrm{~cm}.$ Find $T A, T X$ and radius of the circle. $\text{ (5 marks)}$

12.(figure)In the figure $\angle B A C=40^{\circ}, \angle A F D=85^{\circ}$ and $\angle F D C=30^{\circ}$ Prove that $B C E F$ is a cyclic quadrilateral. $\text{ (5 marks)}$


13.(figure)In the figure, $\angle E A C=40^{\circ}, \angle A E D=80^{\circ}$, $\angle C D E=20^{\circ}.$ Prove that $B, C, F, E$ are concyclic.$\angle C D E=$ $A$. $\text{ (5 marks)}$


14.(figure)In the figure, $\angle E A F=45^{\circ}, \angle A E D=80^{\circ}$, $\angle C D E=25^{\circ}.$ Prove that $B, C, F, E$ are concyclic. $\text{ (5 marks)}$


15.$A B$ is a diameter of the circle $A P B.$ A line perpendicular to $A B$ intersects $A B$, $A P$ at $H, K$ respectively.Prove that $K H B P$ is cyclic. $\text{ (5 marks)}$

16.$P Q R S$ is a parallelogram.A circle through $P, Q$ cuts the diagonals $P R, Q S$ at $A, B$ respectively.Prove that $A, B, \mathrm{~S}, R$ are concyclic. $\text{ (5 marks)}$

17.In circle $O$, the diameter $A B$ is produced to $C$ and the line $C T$ is a tangent to the circle at $T.$ The line drawn perpendicular to $A C$ at $C$ meets $A T$ produced at $D.$ Prove that $B C D T$ is cyclic quadrilateral and $C T=C D.$ $\text{ (5 marks)}$

18.Prove that the quadrilateral formed by producing the bisectors of the interior angles of any quadrilateral is cyclic. $\text{ (5 marks)}$

19.(figure)In the figure, $O$ is the centre of the circle.The diameter $A B$ is produced to $C$ and the line $C T$ is the tangent to the circle at $T.$ The line drawn perpendicular to $A C$, at $C$, meets $A T$ produced at $D.$ Prove (i) $B C D T$ is a cyclic quadrilateral. $\text{ (5 marks)}$


20.Two circles $P$ and $Q$ intersect at $A$ and $D.$ A line $B D C$ is drawn to cut the circle $P$ at $B$ and the circle $Q$ at $C.$ If $A P D Q$ is a cyclic quadrilateral, prove that $B C^{2}=A B^{2}+A C^{2}$. $\text{ (5 marks)}$



Answer (2010)


$\quad\;\,$$\,$
1.Prove
2.Prove
3.Prove
4.Prove
5.Prove
6.Prove
7.Prove
8.$120^{\circ}$
9.Prove
10.Prove
11.TA=3 cm,TX=5 cm, radius=1.5 cm
12.Prove
13.Prove
14.Prove
15.Prove
16.Prove
17.Prove
18.Prove
19.Prove
20.Prove

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