Line Segment That Subtends Equal Angles at Two Other Points

Q.

In the given figure, PQ = QR, $\angle RQP={68}^{\circ }$, PC and CQ are tangents to the circle with center O. Calculate the values of:

$\left(i\right)\angle QOP$

$\left(ii\right)\angle QCP$ [4 MARKS]

Q.

Prove that two different circles cannot intersect each other at more than two points.

Q.

$A,B\mathrm{and}C$ are three points on a circle. Prove that the perpendicular bisectors of $AB,BC\mathrm{and}CA$ are concurrent.

Q.

$O$ is the Circumcenter of the triangle$△ABC$ and $D$ is the mid-point of the base $BC$. Prove that $\angle BOD=\angle A$.

Q. In figure given below, O is the centre of circle and $\angle AQR={35}^{\circ }$ and $\angle PAQ={120}^{\circ }$ then $\angle SPR$ is equal to :

1. ${30}^{\circ }$
2. ${35}^{\circ }$
3. ${25}^{\circ }$
4. ${20}^{\circ }$

Q.

In the adjoining figure. POQ is the diameter of the circle. R and S are any two points on the circle. Then,

1. ∠PRQ > ∠PSQ

2. ∠PRQ < ∠PSQ

3. ∠PRQ = ∠PSQ

4. ∠PRQ =1/2 ∠PSQ

Q.

In figure, $\angle$PSR=${100}^{\circ }$, where P, Q and R are points on a circle with centre O.Find $\angle$OPR (in degrees).

__

Q. A circular park of radius $20m$ is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone his hands to talk each other. Find the length of the string of each phone.

Q.

A, B, C, D are 4 non collinear points in a plane such that $\angle ACB$ =$\angle ADB$ , then how many circle(s) can be drawn passing through all 4 points.

1. 0
2. 1
3. 2
4. 3

Q. Two circles of radii $18cm$ and $8cm$ touch externally. The length of a direct common tangent to the two circles is

Q. Find the equations of circles which passes through the vertices of the triangle formed by the line
x + y + 1 = 0 , 3x +y - 5 = 0 and 2x + y - 5 = 0 z

Q.

A, B, C, D are 4 non collinear points in a plane such that $\angle$ACB = $\angle$ADB, then how many circle(s) can be drawn passing through all 4 points?

1. 0

2. 1

3. 2

4. 3

Q. Construct a square inscribed in a circle of radius $3$ cm.

Q.

If arcs $AXB$ and $CYD$ of a circle are congruent, find the ratio of $AB$ and $CD.$

Q. In Figure, $AOB$ is a diameter of the circle, $C,D$ and $E$ are any three points on the semi-circle . Find the value of $\angle ACD+\angle BED$.

Q. In figure, $\angle$PQR$={100}^o$, where P, Q and R are points on a circle with centre O. Find $\angle$OPR.

Q. If $2x-y+4=0$ is a diameter of a circle which circumscribes a rectangle ABCD. If the co-ordinates of $A,B$ are $(4,6)$ and $(1,9)$ , find the area of ABCD , and show that it is a square.

Q.

ABCD is a parallelogram. A circle through A, B is so drawn that it intersects AD at P and BC at Q. Prove that P, Q, C, and D are concyclic.

Q. In figure (not to scale), the points M, R, N, S, Q are concyclic. Find $\angle PQR+\angle OPR+\angle NMS+\angle OSN$, if 'O' is centre of circle.

Q. PQRS is a cyclic quadrilateral, in which if $\angle P-\angle R={30}^o$ and $\angle Q-\angle S=-{30}^o$, then PQRS is a

1. parallelogram
2. square
3. rhombus
4. trapezium

Q. Suppose $AB=BC$, $\angle ABC={68}^o$ , $DA$ and $DB$ are the tangents to the circle with centre O, for the trianlge $ABC$
Calculate the measure of $\angle ACB$

Q. The radius of the circle passing through the vertices of the triangle $ABC$, is

Q. Tangent at $P$ to the circumcircle of triangle $PQR$ is drawn. If this tangent is parallel to side $QR$ show that $△PQR$ is an isosceles triangle.

Q. Seg $PM$ and Seg $PN$ are congruent chords of a circle with centre $C$. Show that the ray $PC$ is the bisector of $\angle NPM$

Q. In the given figure, $BD=DC$ and $\angle DBC={30}^{\circ }$ what is the value of $\angle BAC$?

1. ${40}^{\circ }$
2. ${50}^{\circ }$
3. ${65}^{\circ }$
4. ${60}^{\circ }$

Q. Prove that two different circles cannot intersect each other at more than two points.

Q. Prove that the equation of the circle circumscribing the triangle formed by the sides whose equations are 2x + y - 3 = 0 , 3x - y - 7 = 0 and x - 2y + 1 = 0 is $x^2+y^2-$ 5x - y + 4 = 0