Circle

A circle is a geometrical figure in a plane consisting of points that are—

Concepts like centre, diameter, radius and chord were discussed in—

Which topics are introduced specifically in Chapter 8?

By the end of the chapter, students will be able to do all of the following EXCEPT—

The fixed point from which all points of a circle are equidistant is called the—

The closed path traced by a point that keeps a fixed distance from the centre is called a—

The distance from the centre to any point on a circle is called the—

If $O$ is a fixed point and $r$ a fixed measurement, the set of points at distance $r$ from $O$ forms—

In the figure where $O$ is the centre and $A,B,C$ are on the circle, each of $OA,OB,OC$ is a—

Some coplanar points are called concyclic if—

Three points $A,B,C$ on a circle are—

The minimum number of points required to determine a unique circle is—

If three points are collinear, then a circle passing through all three—

The locus of all points in a plane equidistant from a fixed point is—

The set of all points in the plane whose distance from the centre is less than the radius is the—

The set of all points whose distance from the centre is greater than the radius is the—

The line segment joining two points of a circle lies—

A line segment from an interior point to an exterior point of a circle intersects the circle at—

If $P$ is interior and $Q$ is exterior to a circle, then segment $PQ$ meets the circle at—

A point lies on the circle if its distance from the centre is—

The line segment connecting two different points of a circle is called a—

A chord that passes through the centre of the circle is called the—

The centre of a circle is the midpoint of any—

If the radius is $r$, then the length of a diameter is—

Which statement about a diameter is true?

Every diameter of a circle is also a—

A radius and a diameter of the same circle are related by—

If the diameter of a circle is $10$ cm, the radius is—

The line segment drawn from the centre of a circle to bisect a chord (other than diameter) is—

In Theorem 17, $\triangle OAM \cong \triangle OBM$ is established by which congruence rule?

In the proof of Theorem 17, which sides are taken as equal in $\triangle OAM$ and $\triangle OBM$?

After proving $\angle OMA = \angle OMB$ and noting they form a straight angle, each must be—

Corollary 1 of Theorem 17 states that the perpendicular bisector of any chord—

Corollary 2 of Theorem 17 states that a straight line cannot intersect a circle in more than—

The converse stated in the Work box of Theorem 17 says: the perpendicular from the centre of a circle to a chord—

If $M$ is the midpoint of chord $AB$ and $O$ is the centre, then $OM \perp AB$ holds for—

From the centre $O$, a perpendicular dropped on a chord of length $8$ cm meets the chord at distance—

The Work box challenges the student to prove that the perpendicular from the centre—

Theorem 18 states that all equal chords of a circle are—

In Theorem 18, $OE$ and $OF$ are drawn—

Because the perpendicular from the centre bisects the chord, $AE$ equals—

In the proof of Theorem 18, which congruence rule is used to show $\triangle OAE \cong \triangle OCF$?

After $\triangle OAE \cong \triangle OCF$, we conclude—

The hypotenuse used in Theorem 18's RHS step is—

In Theorem 18, $OA = OC$ because they are—

If two chords of a circle are equal, then the perpendicular distances from the centre to those chords are—

Theorem 19 states that chords equidistant from the centre of a circle are—

Theorem 19 is the converse of—

In Theorem 19, $OE = OF$ implies that—

In Theorem 19's proof, $\triangle OAE \cong \triangle OCF$ is justified by which rule?

From $AE = CF$ and the fact that perpendicular from centre bisects chord, we conclude—

Corollary 3 (following Theorem 19) states that—

Of two chords of a circle, the one closer to the centre is—

The greatest chord of a circle has length—

The straight line joining the midpoints of two parallel chords of a circle—

If two chords $AB$ and $AC$ of a circle subtend equal angles with the radius through $A$, then—

If a circle passes through the vertices of a right-angled triangle, then the centre of the circle is the—

In two concentric circles a chord $AB$ of the outer circle meets the inner circle at $C$ and $D$. Then—

If two equal chords of a circle intersect, the segments of one chord—

Two equal chords drawn from the two ends of a diameter on its opposite sides are—

Of two chords of a circle, the bigger chord is—

In the figure with $PQ = x$ cm and $OR \perp PQ$, $\angle QOS$ equals—

In the same figure, the largest chord of the circle is—

A chord $PQ = x$ cm of a circle with centre $O$ has $OR \perp PQ$. If $OR = (x-2)$ cm and the radius equals the diameter logic gives $x$ cm such that $x^2 = (x-2)^2 + \left(\dfrac{x}{2}\right)^2$—the value of $x$ is—

If a line joining two points makes equal angles at two different points on the same side of the line, then the four points are—

The midpoints of equal chords of a circle are—

Two parallel chords drawn from the two ends of a diameter on its opposite sides are—

If two chords of a circle bisect each other, their point of intersection is—

The piece of a circle between any two points of the circle is called—

Two distinct points on a circle divide it into—

The larger of the two arcs is called the—

The smaller of the two arcs is called the—

The terminal points of an arc are also—

An arc with internal point $R$ between $A$ and $B$ is denoted by—

The two arcs determined by points $A$ and $B$ on a circle have—

Often the minor arc between $A$ and $B$ is denoted simply by—

An angle is said to cut an arc of a circle if—

For an angle to cut an arc, each side of the angle must contain—

For an angle to cut an arc, every interior point of the arc must lie—

In the standard figure, the angle cuts arc—

An angle that cuts an arc of a circle determines the arc by intersecting—

If two chords of a circle meet at a point on the circle, the angle between them is called—

The vertex of an inscribed angle lies—

The inscribed angle "stands on" which arc?

An inscribed angle is also said to be inscribed in the—

In the figure, the angle stands on arc $APB$ and is inscribed in—

Arcs $APB$ and $ACB$ in the same circle are mutually—

An angle whose vertex is at the centre of the circle is called—

The vertex of a central angle always lies at—

The sides of a central angle pass through—

Every central angle (except a straight angle) stands on a—

In a semicircle, the angle at the centre $\angle BOC$ is a—

In a semicircle, the angle on the arc $\angle BAC$ at any point on the circle is a—

Theorem 20: the angle subtended by an arc at the centre is— the angle subtended by the same arc at any point on the remaining part of the circle.

If an arc subtends an angle of $80^{\circ}$ at the centre, the angle subtended at the circumference is—

If an inscribed angle is $35^{\circ}$, the corresponding central angle is—

In the proof of Theorem 20, the auxiliary line drawn through $A$ passes through—

The proof uses the fact that an exterior angle of a triangle equals—

In $\triangle AOB$, $OA = OB$ implies $\angle BAO = \angle ABO$ because—

Since $\angle BOD = \angle BAO + \angle ABO = 2\angle BAO$, similarly $\angle COD$ equals—

Adding the two relations gives $\angle BOC = \dotsb$

The Work box accompanying Theorem 20 asks to prove the theorem when—

The angle standing on an arc of the circle equals— the angle subtended by the arc at the centre.

Angles in a circle standing on the same arc are—

In Theorem 21, $\angle BAD$ and $\angle BED$ standing on arc $BCD$ are equal because each equals—

If two angles inscribed in a circle subtend the same chord and lie on the same side of it, they are—

The fact "angles in the same segment are equal" is a direct consequence of—

In Theorem 21, the central angle subtended by arc $BCD$ is referenced to convert two inscribed angles into—

The angle inscribed in a semicircle is—

In Theorem 22, the central angle $\angle AOB$ on diameter $AB$ equals—

Since $\angle ACB = \frac{1}{2}\angle AOB$ and $\angle AOB$ is a straight angle, $\angle ACB$ equals—

Corollary 4 of Theorem 22 says: the circle drawn with the hypotenuse of a right-angled triangle as diameter—

Corollary 5 states the angle inscribed in a major arc is—

The Work box following Corollary 5 asks to prove that any angle inscribed in a minor arc is—

If $AB$ is a diameter of a circle and $C$ is any point on the circle (other than $A,B$), then $\angle ACB$ equals—

The locus of points $P$ such that $\angle APB = 90^{\circ}$ for a fixed segment $AB$ is—

$ABCD$ is a cyclic quadrilateral with diagonals $AC,BD$ meeting at $E$. Then $\angle AOB + \angle COD$ equals—

In a cyclic quadrilateral $ABCD$, if $\angle ADB + \angle BDC = 1$ right angle, then which three points are collinear?

The oblique sides of a cyclic trapezium are—

In a circle with centre $O$ and $OB = 2.5$ cm, the perimeter of the (cyclic) figure $ABCD$ depends on—

In Exercise 8.2 Q5, two chords $AB$ and $CD$ intersect at $E$. Then $\triangle AED$ and $\triangle BEC$ are—

A quadrilateral having all four vertices on a circle is called—

Theorem 23 states that the sum of the two opposite angles of a cyclic quadrilateral is—

In a cyclic quadrilateral $ABCD$, $\angle ABC + \angle ADC$ equals—

In a cyclic quadrilateral $ABCD$, $\angle BAD + \angle BCD$ equals—

The proof of Theorem 23 uses—

Reflex $\angle AOC + \angle AOC$ equals—

Corollary 6: if one side of a cyclic quadrilateral is extended, the exterior angle is—

Corollary 7: a parallelogram inscribed in a circle is a—

If a cyclic quadrilateral has $\angle A = 70^{\circ}$, then $\angle C$ is—

If $\angle PQR$ of cyclic quadrilateral $PQRS$ is $110^{\circ}$, then $\angle PSR$ is—

Theorem 24 states that if two opposite angles of a quadrilateral are supplementary, then the four vertices are—

Theorem 24 is the converse of—

The proof of Theorem 24 begins by assuming a circle through three of the four vertices and seeking a contradiction with the position of—

In the proof, $E$ is the second intersection point of side $AD$ with the auxiliary circle through $A,B,C$. We deduce $\angle AEC = \angle ADC$, which contradicts—

The conclusion of Theorem 24 is that—

The internal and external bisectors of $\angle B$ and $\angle C$ of $\triangle ABC$ meeting at $P$ and $Q$ make $B,P,C,Q$—

In Exercise 8.3 Q2, four points $A,Q,P,B$ are—

If chords $AB,CD$ of a circle with centre $O$ meet at right angles inside the circle, then $\angle AOD + \angle BOC$ equals—

In a cyclic quadrilateral $ABCD$ with opposite angles supplementary and $AC$ bisecting $\angle BAD$, we have—

In a circle of radius $2.5$ cm with $AB = 3$ cm and $BD$ bisecting $\angle ADC$, the length $AD$ equals—

If two triangles standing on equal bases have supplementary vertical angles, their circumcircles are—

The bisector of an angle of a cyclic quadrilateral and the exterior bisector of the opposite angle meet—

A straight line and a circle in a plane can have at most—

If a straight line intersects a circle at exactly two points, the line is called—

If a straight line touches a circle at exactly one point, the line is called—

The single common point of a tangent and a circle is called—

All points of a secant lying between the two points of intersection lie—

The number of tangents that can be drawn to a circle from a point on the circle is—

The number of tangents that can be drawn to a circle from a point inside the circle is—

The number of tangents that can be drawn to a circle from an external point is—

A line that is tangent to two circles at once is called a—

If the points of contact of a common tangent are different and the centres lie on the same side of the tangent, the tangent is called—

If the centres of two circles lie on opposite sides of a common tangent (with distinct points of contact), the tangent is—

If a common tangent touches both circles at the same point, the two circles—

Two circles touch externally when their centres lie—

Two circles touch internally when their centres lie—

If two circles intersect each other externally and one diameter is $8$ cm while the other radius is $4$ cm, the distance between their centres is—

The maximum number of common tangents to two circles that intersect each other in two points is—