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CIRCLES - Q&A

Exercise 9.1

1. Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.

Given: Two congruent circles with centres O and O' and radii r. AB is a chord of the first circle and CD is a chord of the second circle such that AB = CD.
To Prove: ∠ AOB = ∠ CO'D.

Proof:
In Δ AOB and Δ CO'D:
1. AB = CD (Given)
2. OA = O'C (Radii of congruent circles)
3. OB = O'D (Radii of congruent circles)

Therefore, by SSS congruence rule:
Δ AOB ≅ Δ CO'D.

By CPCT (Corresponding Parts of Congruent Triangles):
∠ AOB = ∠ CO'D.
Hence, equal chords of congruent circles subtend equal angles at their centres.

2. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

Given: Two congruent circles with centres O and O'. ∠ AOB = ∠ CO'D.
To Prove: Chord AB = Chord CD.

Proof:
In Δ AOB and Δ CO'D:
1. OA = O'C (Radii of congruent circles)
2. ∠ AOB = ∠ CO'D (Given)
3. OB = O'D (Radii of congruent circles)

Therefore, by SAS congruence rule:
Δ AOB ≅ Δ CO'D.

By CPCT:
AB = CD.
Hence, the chords are equal.


Exercise 9.2

1. Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.

Let the two circles be with centres O and O' and radii 5 cm and 3 cm respectively.
Let them intersect at points A and B. So AB is the common chord.
The distance between centres OO' = 4 cm.

Let OO' intersect AB at M.
We know that the line joining centres is the perpendicular bisector of the common chord.
So, AB ⊥ OO' and M is the mid-point of AB.

Let OM = x cm. Then O'M = (4 - x) cm.
In right-angled Δ OMA:
OA2 = OM2 + AM2
52 = x2 + AM2
AM2 = 25 - x2 ... (i)

In right-angled Δ O'MA:
O'A2 = O'M2 + AM2
32 = (4 - x)2 + AM2
AM2 = 9 - (4 - x)2 ... (ii)

From (i) and (ii):
25 - x2 = 9 - (16 + x2 - 8x)
25 - x2 = 9 - 16 - x2 + 8x
25 = -7 + 8x
32 = 8x ⇒ x = 4.

Since x = 4, the centre O' coincides with M (because OO' = 4).
Substitute x = 4 in (i):
AM2 = 25 - 42 = 25 - 16 = 9.
AM = 3 cm.

Length of common chord AB = 2 × AM = 2 × 3 = 6 cm.
(Note: Since radius of smaller circle is 3 cm and distance between centres is 4 cm and radius of larger is 5 cm, the common chord passes through the centre of the smaller circle).

2. If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.

Given: A circle with centre O. Equal chords AB and CD intersect at P.
To Prove: AP = CP and BP = DP.

Construction: Draw OM ⊥ AB and ON ⊥ CD. Join OP.

Proof:
In right-angled Δ OMP and Δ ONP:
1. OP = OP (Common hypotenuse)
2. OM = ON (Equal chords are equidistant from centre)
3. ∠ OMP = ∠ ONP = 90°
So, Δ OMP ≅ Δ ONP (by RHS).
Therefore, MP = NP (by CPCT).

Since AB = CD, their halves are also equal.
AM = CN ... (i)
Also, MP = NP ... (ii)

Adding (i) and (ii):
AM + MP = CN + NP
AP = CP.

Subtracting AP = CP from AB = CD:
AB - AP = CD - CP
BP = DP.
Hence Proved.

3. If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.

Given: Equal chords AB and CD intersect at P. O is the centre.
To Prove: ∠ OPA = ∠ OPC (or ∠ OPE = ∠ OPF if we take perpendiculars).

Construction: Draw OM ⊥ AB and ON ⊥ CD. Join OP.

Proof:
In right-angled Δ OMP and Δ ONP:
1. OP = OP (Common)
2. OM = ON (Equal chords are equidistant from centre)
3. ∠ OMP = ∠ ONP = 90°
So, Δ OMP ≅ Δ ONP (by RHS).

Therefore, ∠ OPM = ∠ OPN (by CPCT).
This means the line OP makes equal angles with the chords AB and CD.

4. If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD (see Fig. 9.12).

Given: A line intersects two concentric circles at A, B, C, D.
To Prove: AB = CD.

Construction: Draw OM ⊥ AD.

Proof:
For the outer circle, AD is a chord and OM ⊥ AD.
Perpendicular from centre bisects the chord.
So, AM = DM ... (i)

For the inner circle, BC is a chord and OM ⊥ BC.
So, BM = CM ... (ii)

Subtracting (ii) from (i):
AM - BM = DM - CM
AB = CD.
Hence Proved.

5. Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius 5m drawn in a park. Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is 6m each, what is the distance between Reshma and Mandip?

Let positions of Reshma, Salma, Mandip be R, S, M.
RS = 6m, SM = 6m. Radius OS = 5m, OR = 5m, OM = 5m.
Since RS = SM, quadrilateral ORSM is a kite-like structure (adjacent sides equal).
But actually, R, S, M are on a circle.
Since RS = SM, equal chords subtend equal angles at centre. ∠ ROS = ∠ SOM.
So, OS bisects ∠ ROM. Therefore OS ⊥ RM and bisects RM.
Let OS intersect RM at K.

In Δ ORS, sides are 5, 5, 6.
Semi-perimeter s = (5+5+6)/2 = 8.
Area of Δ ORS = √[s(s-a)(s-b)(s-c)] = √[8(3)(3)(2)] = √144 = 12 m2.

Also Area of Δ ORS = 1/2 × Base × Height = 1/2 × OS × RK.
12 = 1/2 × 5 × RK
24 = 5 × RK
RK = 4.8 m.

Since OS bisects RM, RM = 2 × RK.
RM = 2 × 4.8 = 9.6 m.
Distance between Reshma and Mandip is 9.6 m.

6. 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 in his hands to talk each other. Find the length of the string of each phone.

Let Ankur, Syed, David be at A, S, D.
Since they are at equal distances, AS = SD = DA.
So Δ ASD is an equilateral triangle. Let side be 'a'.
Radius circumscribing the equilateral triangle R = 20m.

We know relation between side of equilateral triangle and circumradius:
R = a / √3
20 = a / √3
a = 20√3 m.

So, the length of the string of each phone is 20√3 m.


Exercise 9.3

1. In Fig. 9.23, A, B and C are three points on a circle with centre O such that ∠ BOC = 30° and ∠ AOB = 60°. If D is a point on the circle other than the arc ABC, find ∠ ADC.

∠ AOC = ∠ AOB + ∠ BOC
∠ AOC = 60° + 30° = 90°.

The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
So, ∠ AOC = 2 × ∠ ADC.
90° = 2 × ∠ ADC
∠ ADC = 45°.

2. A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

Let chord be AB and centre be O. Radius OA = OB = r.
Given Chord AB = r.
So, Δ OAB is an equilateral triangle.
Therefore, angle at centre ∠ AOB = 60°.

Angle at major arc (say at point P):
∠ APB = 1/2 ∠ AOB = 1/2 (60°) = 30°.

Angle at minor arc (say at point Q):
AQBP is a cyclic quadrilateral.
∠ APB + ∠ AQB = 180° (Opposite angles sum to 180°)
30° + ∠ AQB = 180°
∠ AQB = 150°.

Answer: Angle at major arc = 30°, Angle at minor arc = 150°.

3. In Fig. 9.24, ∠ PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠ OPR.

Take a point S on the major arc.
PQRS is a cyclic quadrilateral.
∠ PQR + ∠ PSR = 180°
100° + ∠ PSR = 180°
∠ PSR = 80°.

Angle at centre ∠ POR = 2 × ∠ PSR = 2 × 80° = 160°.

In Δ OPR, OP = OR (Radii).
So, ∠ OPR = ∠ ORP.
Sum of angles = 180°
∠ OPR + ∠ ORP + ∠ POR = 180°
2 ∠ OPR + 160° = 180°
2 ∠ OPR = 20°
∠ OPR = 10°.

4. In Fig. 9.25, ∠ ABC = 69°, ∠ ACB = 31°, find ∠ BDC.

In Δ ABC:
∠ BAC + ∠ ABC + ∠ ACB = 180°
∠ BAC + 69° + 31° = 180°
∠ BAC + 100° = 180°
∠ BAC = 80°.

Angles in the same segment are equal.
Therefore, ∠ BDC = ∠ BAC.
∠ BDC = 80°.

5. In Fig. 9.26, A, B, C and D are four points on a circle. AC and BD intersect at a point E such that ∠ BEC = 130° and ∠ ECD = 20°. Find ∠ BAC.

Step 1: Find ∠ CED and ∠ EDC.
∠ CED + ∠ BEC = 180° (Linear Pair)
∠ CED + 130° = 180°
∠ CED = 50°.

In Δ CDE:
∠ CDE + ∠ DCE + ∠ CED = 180°
∠ CDE + 20° + 50° = 180°
∠ CDE = 110°.
(Note: ∠ CDE is same as ∠ CDB).

Step 2: Find ∠ BAC.
Angles in the same segment are equal.
∠ BAC = ∠ BDC (Subtended by arc BC).
Since ∠ BDC = ∠ CDE = 110°.
So, ∠ BAC = 110°.

6. ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠ DBC = 70°, ∠ BAC is 30°, find ∠ BCD. Further, if AB = BC, find ∠ ECD.

Step 1: Find ∠ BDC.
Angles in the same segment are equal.
∠ BDC = ∠ BAC.
Since ∠ BAC = 30°, ∠ BDC = 30°.

Step 2: Find ∠ BCD.
In Δ BCD:
∠ DBC + ∠ BDC + ∠ BCD = 180°
70° + 30° + ∠ BCD = 180°
100° + ∠ BCD = 180°
∠ BCD = 80°.

Step 3: Find ∠ ECD if AB = BC.
If AB = BC, then in Δ ABC, ∠ BAC = ∠ BCA.
So, ∠ BCA = 30°.

We know ∠ BCD = 80°.
∠ BCD = ∠ BCA + ∠ ACD (or ∠ ECD)
80° = 30° + ∠ ECD
∠ ECD = 50°.

7. If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

Let ABCD be the cyclic quadrilateral and AC, BD be diagonals which are diameters.
Since AC is a diameter, angle in a semi-circle is 90°.
So, ∠ ABC = 90° and ∠ ADC = 90°.
Since BD is a diameter:
∠ BAD = 90° and ∠ BCD = 90°.
Since all angles of the quadrilateral are 90°, ABCD is a rectangle.

8. If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

Given: Trapezium ABCD with AB || CD and AD = BC.
To Prove: ABCD is cyclic (i.e., ∠ A + ∠ C = 180°).

Construction: Draw DE ⊥ AB and CF ⊥ AB.

Proof:
In right Δ DEA and Δ CFB:
1. AD = BC (Given)
2. DE = CF (Distance between parallel lines)
3. ∠ DEA = ∠ CFB = 90°
So, Δ DEA ≅ Δ CFB (by RHS).
Therefore, ∠ A = ∠ B (by CPCT).

Since AB || CD, consecutive interior angles sum to 180°.
∠ A + ∠ D = 180°
∠ B + ∠ C = 180°

Substitute ∠ B with ∠ A in the second equation:
∠ A + ∠ C = 180°.
Since opposite angles sum to 180°, ABCD is a cyclic quadrilateral.

9. Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see Fig. 9.27). Prove that ∠ ACP = ∠ QCD.

Proof:
For the first circle (left side):
Angles in the same segment are equal.
∠ ACP = ∠ ABP (Subtended by arc AP). ... (i)

For the second circle (right side):
∠ QCD = ∠ QBD (Subtended by arc QD). ... (ii)

But ∠ ABP = ∠ QBD (Vertically opposite angles). ... (iii)

From (i), (ii), and (iii):
∠ ACP = ∠ QCD.
Hence Proved.

10. If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lies on the third side.

Given: Δ ABC. Two circles with diameters AB and AC intersect at A and D.
To Prove: D lies on BC.

Proof:
Join AD.
Since AB is a diameter, ∠ ADB is an angle in a semicircle.
So, ∠ ADB = 90°.
Since AC is a diameter, ∠ ADC is an angle in a semicircle.
So, ∠ ADC = 90°.

Adding angles:
∠ ADB + ∠ ADC = 90° + 90° = 180°.
This implies that BDC is a straight line.
Therefore, D lies on the line segment BC.

11. ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠ CAD = ∠ CBD.

Given: ∠ ABC = 90° and ∠ ADC = 90°.

Proof:
Since ∠ ABC + ∠ ADC = 90° + 90° = 180°, the quadrilateral ABCD is cyclic (Sum of opposite angles is 180°).
Consider the circle passing through A, B, C, D.
Angles in the same segment are equal.
∠ CAD and ∠ CBD are in the same segment (subtended by arc CD).
Therefore, ∠ CAD = ∠ CBD.
Hence Proved.

12. Prove that a cyclic parallelogram is a rectangle.

Given: ABCD is a cyclic parallelogram.
To Prove: ABCD is a rectangle.

Proof:
Since ABCD is a parallelogram, opposite angles are equal.
∠ A = ∠ C.
Since ABCD is cyclic, sum of opposite angles is 180°.
∠ A + ∠ C = 180°.

Substitute ∠ C with ∠ A:
2 ∠ A = 180°
∠ A = 90°.

Since a parallelogram with one angle 90° is a rectangle, ABCD is a rectangle.

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Quick Review Flashcards - Click to flip and test your knowledge!
Question
If a line segment $PQ$ and a point $R$ (not on the line) are joined to form $\angle PRQ$, what is this angle called in relation to $PQ$?
Answer
The angle subtended by the line segment $PQ$ at the point $R$.
Question
In a circle with center $O$ and chord $PQ$, what is the specific name for $\angle POQ$?
Answer
The angle subtended by the chord $PQ$ at the center.
Question
If a chord $PQ$ subtends $\angle PRQ$ and $\angle PSQ$ at points on the major and minor arcs respectively, what are these angles called?
Answer
Angles subtended by the chord $PQ$ at points on the circle.
Question
What is the relationship between the length of a chord and the size of the angle it subtends at the center of a circle?
Answer
The longer the chord, the bigger the angle subtended at the center.
Question
Theorem 9.1: Equal chords of a circle subtend _____ angles at the center.
Answer
equal
Question
What congruence rule is typically used to prove that equal chords subtend equal angles at the center of a circle?
Answer
SSS rule (Side-Side-Side rule).
Question
In the proof of Theorem 9.1, why are $OA$ and $OC$ considered equal in $\triangle AOB$ and $\triangle COD$?
Answer
They are radii of the same circle.
Question
What does the abbreviation 'CPCT' stand for in geometric proofs?
Answer
Corresponding parts of congruent triangles.
Question
Theorem 9.2: If the angles subtended by the chords of a circle at the center are equal, then the chords are _____.
Answer
equal
Question
Theorem 9.2 is considered the _____ of Theorem 9.1.
Answer
converse
Question
How many circles are considered congruent if they have the same radii?
Answer
Two (or more).
Question
True or False: Equal chords of congruent circles subtend equal angles at their respective centers.
Answer
True
Question
Theorem 9.3: The perpendicular from the center of a circle to a chord _____ the chord.
Answer
bisects
Question
According to Theorem 9.4, if a line is drawn through the center of a circle to bisect a chord, what is its relationship to that chord?
Answer
It is perpendicular to the chord.
Question
In the proof of Theorem 9.4, what geometric property ensures $\triangle OAM \cong \triangle OBM$ given $M$ is the midpoint and $O$ is the center?
Answer
SSS rule (Side-Side-Side rule).
Question
How is the distance from a point to a line mathematically defined?
Answer
The length of the perpendicular from the point to the line.
Question
Of all line segments joining a point $P$ to a line $AB$, which segment has the least length?
Answer
The perpendicular from $P$ to $AB$.
Question
If a point lies directly on a line, what is the distance between the point and the line?
Answer
Zero
Question
What is the distance of a diameter from the center of its circle?
Answer
Zero
Question
Theorem 9.5: Equal chords of a circle are _____ from the center.
Answer
equidistant
Question
Theorem 9.6: Chords that are equidistant from the center of a circle are _____ in length.
Answer
equal
Question
In a circle, a longer chord is _____ to the center than a smaller chord.
Answer
nearer
Question
What is the longest chord in any circle?
Answer
The diameter.
Question
If two chords of a circle are equal, their corresponding arcs are _____.
Answer
congruent
Question
If two arcs of a circle are congruent, then their corresponding chords are _____.
Answer
equal
Question
How is the angle subtended by a major arc at the center defined?
Answer
As the reflex angle subtended by the arc at the center.
Question
True or False: Congruent arcs of a circle subtend equal angles at the center.
Answer
True
Question
Theorem 9.7: The angle subtended by an arc at the center is _____ the angle subtended by it at any point on the remaining part of the circle.
Answer
double
Question
According to Theorem 9.7, if $\angle PAQ$ is subtended by arc $PQ$ at the circumference, what is the measure of the central angle $\angle POQ$?
Answer
$2 \angle PAQ$
Question
When using Theorem 9.7 for a major arc $PQ$, what specific central angle is used in the equation $Central\ Angle = 2 \angle PAQ$?
Answer
Reflex $\angle POQ$
Question
Theorem 9.8: Angles in the same _____ of a circle are equal.
Answer
segment
Question
What is the measure of an angle in a semicircle?
Answer
$90^{\circ}$
Question
Theorem 9.9: If a line segment joining two points subtends equal angles at two other points on the same side, then all four points are _____.
Answer
concyclic
Question
Definition: A quadrilateral $ABCD$ where all four vertices lie on a circle.
Answer
Cyclic quadrilateral
Question
Theorem 9.10: What is the sum of either pair of opposite angles of a cyclic quadrilateral?
Answer
$180^{\circ}$
Question
Theorem 9.11: If the sum of a pair of opposite angles of a quadrilateral is $180^{\circ}$, then the quadrilateral is _____.
Answer
cyclic
Question
In a circle, if a chord is equal to the radius, what is the measure of the angle it subtends at the center?
Answer
$60^{\circ}$
Question
If two equal chords intersect within a circle, what can be said about the segments of the chords?
Answer
The segments of one chord are equal to the corresponding segments of the other.
Question
If a line intersects two concentric circles at points $A, B, C,$ and $D$ in order, what is the relationship between $AB$ and $CD$?
Answer
$AB = CD$
Question
If the diagonals of a cyclic quadrilateral are diameters of the circle, what kind of quadrilateral is it?
Answer
A rectangle.
Question
If the non-parallel sides of a trapezium are equal, what special property does the trapezium have?
Answer
It is cyclic.
Question
In $\triangle OAQ$, if $OA = OQ$ (radii), what is the relationship between $\angle OAQ$ and $\angle OQA$?
Answer
They are equal (Angles opposite equal sides are equal).
Question
If two circles are drawn using two sides of a triangle as diameters, where does their point of intersection lie?
Answer
On the third side of the triangle.
Question
A cyclic parallelogram is always a _____.
Answer
rectangle
Question
In Example 5, the quadrilateral formed by the internal angle bisectors of any quadrilateral is _____.
Answer
cyclic
Question
What is the sum of the interior angles of any quadrilateral?
Answer
$360^{\circ}$
Question
If $\angle BOC = 30^{\circ}$ and $\angle AOB = 60^{\circ}$ at the center $O$, what is the measure of $\angle ADC$ subtended at the circumference?
Answer
$45^{\circ}$ (Half of $30^{\circ} + 60^{\circ}$)
Question
How is the distance between two intersecting chords measured from the center if the chords are of equal length?
Answer
They are at an equal distance from the center.
Question
What geometric figure is formed by the set of all points in a plane equidistant from a fixed point?
Answer
A circle.
Question
Which theorem states that angles in the same segment are equal?
Answer
Theorem 9.8
Question
In the proof of Theorem 9.7, what triangle property relates the exterior angle to the interior angles?
Answer
An exterior angle is equal to the sum of the two interior opposite angles.
Question
If a triangle is formed by the center and a chord equal to the radius, what specific type of triangle is it?
Answer
An equilateral triangle.
Question
In Theorem 9.7, case (ii), when the arc $PQ$ is a semicircle, what is the value of $\angle POQ$?
Answer
$180^{\circ}$
Question
True or False: A circle can have only a finite number of chords.
Answer
False (It can have infinitely many).
Question
If a line segment $AB$ subtends $\angle ACB = \angle ADB$ where $C$ and $D$ are on the same side, what does this imply about points $A, B, C,$ and $D$?
Answer
They are concyclic (they lie on the same circle).