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Maths 2 Marks Questions

P-2 Circle · Maharashtra Board STD 10 (MSBSHSE - Semi English Medium). 62 questions.

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Question 12 Marks
In figure 3.80, if PQ = 6, QR = 10, PS = 8 find TS.
Answer
Given: PQ = 6, QR = 10, PS = 8
PT × PS = PR × PQ
This property is known as theorem of chords intersecting outside the circle.
⇒ PR = PQ + RQ = 6 + 10 = 16
⇒ PT × 8 = 16 × 6
⇒ PT = 12
TS = PT – PS = 12 – 8 = 4
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Question 22 Marks
In figure 3.78, chord MN and chord RS intersect at point D.
(1) If RD = 15, DS = 4, MD = 8 find DN
(2) If RS = 18, MD = 9, DN = 8 find DS
Answer
Given RD = 15, DS = 4, MD = 8
MD × DN = RD × DS
This property is known as theorem of chords intersecting inside the circle.
⇒ 8 × DN = 15 × 4
$\Rightarrow DN = \frac{15}{2} = 7.5$
(2)Given RS = 18, MD = 9, DN = 8
Here, RS = 18
Let RD = x and DS = 18 – x
MD × DN = RD × DS
This property is known as theorem of chords intersecting inside the circle.
$\Rightarrow 8 \times 9=x \times(18-x)$
$\Rightarrow 18 x-x^2=72$
$\Rightarrow x^2-18 x+72=0$
$\Rightarrow(x-12)(x-6)=0$
$\Rightarrow x=12 \text { or } 6$
$\Rightarrow D S=6 \text { or } 12$
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Question 32 Marks
In figure 3.77, ray PQ touches the circle at point Q. PQ = 12, PR = 8, find PS and RS.
Answer
$\text { Given } P Q=12, P R=8$
$S P \times R P=P Q^2$
This property is known as tangent secant segments theorem.
$\Rightarrow P S \times 8=12^2$
$\Rightarrow PS = \frac{144}{8}= 18$
RS = PS – RP = 18 – 8 = 10
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Question 42 Marks
In figure 3.61, chords AC and DE intersect at B. If ∠ ABE = 108°, m(arc AE) = 95°, find m(arc DC).
Answer
Given ∠ ABE = 108°, m(arc AE) = 95°
Using the property of the secant,
$\angle A B E=\frac{1}{2}[m(\operatorname{arc} A E)+m(\operatorname{arc} D C)] $
$ \Rightarrow 108^{\circ}=\frac{1}{2}\left[95^{\circ}+m(\operatorname{arc} D C)\right]$
⇒m(arc DC) = 108° × 2 – 95°
⇒m(arc DC) = 121°
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Question 52 Marks
In figure 3.60, m(arc NS) = 125°, m(arc EF) = 37°, find the measure ∠ NMS.
Answer
Given m(arc NS) = 125°, m(arc EF) = 37°
Also, ∠ NMS is an external angle, so
$\angle N M S=\frac{1}{2}[m(\operatorname{arcNS})-m(\operatorname{arcEF})] $
$ \Rightarrow \angle N M S=\frac{1}{2}\left[125^{\circ}-37^{\circ}\right]$
$ \Rightarrow \angle N M S=\frac{1}{2} \times 88^{\circ}=44^{\circ}$
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Question 62 Marks
In figure 3.59, altitudes YZ and XT of
∆ WXY intersect at P. Prove that,

(1) $\square$WZPT is cyclic.
(2) Points X, Z, T, Y are concyclic.
Answer
In WZPT,
∠ WZP = ∠ WTP = 90° {YZ and XT are the altitudes}
If a pair of opposite angles of a quadrilateral is supplementary, then the
quadrilateral is cyclic.
⇒ WZPT is cyclic.
(2)∵ X, Z,T,Y lie on same circle, ∴ they are concyclic.
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Question 72 Marks
In figure 3.98, seg AB is a diameter of a circle with centre O . The bisector of∠ ACB intersects the circle at point D. Prove that, seg AD ≅seg BD.

Complete the following proof by filling in the blanks.
Proof: Draw seg OD.
$\angle ACB =\square \ldots 90^{\circ} \ldots \ldots \ldots . . \text { angle inscribed in semicircle }$
$\angle DCB =\square \ldots . . .45^{\circ} \ldots CD \text { is the bisector of } \angle C$
$m (\operatorname{arc} DB )=\square . . . .45^{\circ} \ldots \text { inscribed angle theorem }$
$\angle DOB =\square \ldots . . . .90^{\circ} \text { definition of measure of an arc (I) }$
$\operatorname{seg} OA \cong \operatorname{seg} OB \ldots \ldots . . \text { radii of the circle... } \square \text { (II) }$
$\therefore \text { line } OD \text { is } \square \text { of seg } AB . \ldots . . \text { bisector.... From (I) and (II) }$
$\therefore \text { seg } AD \cong \operatorname{seg} BD$
Answer
Proof: Draw seg OD.
∠ ACB = 90° {angle inscribed in semicircle}
∠ DCB = 45° {CD is the bisector of $\angle C _3$}
m(arc DB) = 45° {inscribed angle theorem}
∠ DOB = 90° {definition of measure of an arc} (I)
seg OA ≅seg OB {radii of the circle}(II)
∴line OD is bisector of seg AB From (I) and (II)
∴seg AD ≅seg BD
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Question 82 Marks
In figure 3.95, chord EF || chord GH. Prove that, chord EG ≅ chord FH.
Fill in the blanks and write the proof.
Proof : Draw seg GF.
∠ EFG = ∠ FGH ...Alternate interior angles.......$\square$(I)
∠ EFG = …………90° ………{inscribed angle theorem}(II)
∠ FGH = …………… 90° ………….{inscribed angle theorem} (III)
∴m(arc EG) = ……………90°…………. from (I), (II), (III).
chord EG ≅ chord FH ..........$\square$
Answer
Proof : Draw seg GF.
∠ EFG = ∠ FGH {Alternate interior angles} (I)
∠ EFG = 90°{inscribed angle theorem}(II)
∠ FGH = 90°{inscribed angle theorem} (III)
∴m(arc EG) = 90° from (I), (II), (III).
chord EG ≅ chord FH {Corresponding chords of congruent arcs of a circle (or congruent circles) are congruent}
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Question 92 Marks
In figure 3.92, O is the centre of a circle, chord PQ ≅ chord RS If∠ POR = 70°and (arc RS) = 80°, find -
(1) m(arc PR)
(2) m(arc QS)
(3) m(arc QSR)
Answer
m(arc PR) = ∠POR = 70°
(2) ∠POQ + ∠ QOS + ∠ROS + ∠POR = 360°
As PQ = RS, ∠ POQ = ∠ROS = 80°
⇒ ∠POQ + ∠ QOS + ∠ROS + ∠POR = 360°
⇒ 80 + ∠QOS + 80 + ∠ 70 = 360
⇒ 230 + ∠ QOS = 360
⇒ ∠QOS = 130°
m(arc QS) = ∠QOS = 130°
(3) m(arc QSR) = ∠QOS + ∠ROS = 130 + 80 = 210°
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Question 102 Marks
In figure 3.91, line PR touches the circle at point Q. Answer the following questions with the help of the figure.
(1) What is the sum of ∠ TAQ and∠ TSQ?
(2) Find the angles which are congruent to ∠ AQP.
(3) Which angles are congruent to ∠QTS ?
(4) ∠ TAS = 65°, find the measure of ∠TQS and arc TS.
(5) If ∠AQP = 42°and ∠SQR = 58° find measure of ∠ATS.
Answer
As TAQS is a cyclic quadrilateral,
∠TAQ + ∠TSQ = 180° (Sum of opposite angles of a cyclic quadrilateral is 180° )
(2) ∠ASQ and ∠ATQ
(3) ∠ QAS and ∠SQR
(4) ∠TAS = 65°
∠ TQS = ∠ TAS = 65° (angle by same arc TS in the same sector)
m(arc TS) = ∠TQS + ∠TAS
⇒ m(arc TS) = 65 + 65 = 130°
(5) ∠AQP + ∠AQS + ∠SQR = 180°
⇒ 42 + ∠AQS + 58 = 180
⇒ ∠AQS + 100 = 180
⇒ ∠AQS = 80
∠ AQS + ∠ ATS = 180° (opposite angles of a cyclic quadrilateral)
⇒ 80 + ∠ATS = 180
⇒ ∠ATS = 100°
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Question 112 Marks
In figure 3.89, line l touches the circle with centre O at point P. Q is the mid point of radius OP. RS is a chord through Q such that chords RS || line l. If RS = 12 find the radius of the circle.
Answer
The radius of the circle will bisect the chord RS. Therefore, RQ = QS = 1/2 × 12 = 6
Let the radius of circle be r,
Now, in Δ OQS, we have,
RQ = 6
OR = r
OQ = 1/2 r
Applying Pythagoras theorem, we get,
${l}r^2=\left(\frac{r}{2}\right)^2+(6)^2 $
$ r^2-\frac{r^2}{4}=36 $
$ \frac{3 r^2}{4}=36$
$3 r^2=4 \times 36$
$r^2=4 \times 12=48$
$r=\sqrt{ } 48 \text { units }$
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Question 122 Marks
In figure 3.85, $\square ABOC$ is a parallelogram. It circumscribes the circle with cnetre T. Point E,F, G, H are touching points. If AE = 4.5,EB = 5.5, find AD.
Answer
Given: AE = 4.5, EB = 5.5
Here, AE = AH = 4.5 {tangents from same external point are equal}
EB = BF = 5.5{tangents from same external point are equal}
∵ Opposite sides of a parallelogram are equal
∴ AE = DG and EB = GC
Also, DH = DG = 4.5{tangents from same external point are equal}
And FC = GC = 5.5 {tangents from same external point are equal}
⇒ AD = AH + HD = 10
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Question 132 Marks
In figure 3.84, O is the centre of the circle. Seg AB, seg AC are tangent segments. Radius of the circle isr and l(AB) = r , Prove that, $\square ABOC$is a square.
Answer
Given: AB = r = radius of the circle
Here, AB = AC = r {tangents from the same external point are equal}
And OB = OC = r = radius of the circle.
⇒ ∠ OBA = ∠ OCA = 90° Using tangent-radius theorem which states that a tangent at any point of a circle is perpendicular to the radius at the point of contact.
∵ sides of ABOC are equal and opposite angles are 90° each
Hence, ABOC is a square.
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Question 142 Marks
In figure 3.76, seg PQ is a diameter of a circle with centre O. R is any point on the circle. seg RS⊥seg PQ.
Prove that, SR is the geometric mean of PS and SQ. [That is, SR² = PS ×SQ]
Image
Answer
Write the proof with the help of the following steps.
(1) Draw ray RS. It intersects the circle at T.
(2) Show that RS = TS.
(3) Write a result using theorem of intersection of chords inside the circle.
(4) Using RS = TS complete the proof.
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Question 152 Marks
Prove that, if two lines containing chords of a circle intersect each other outside the circle, then the measure of angle between them is half the difference in measures of the arcs intercepted by the angle.
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Question 162 Marks
A, B, C are any points on the circle with centre O.
(i) Write the names of all arcs formed due to these points.
(ii) If m arc (BC) = 110° and m arc (AB) = 125°, find measures of all remaining arcs.
Answer
(i) Names of arcs - arc AB, arc BC, arc AC, arc ABC, arc ACB, arc BAC
(ii) m(arc ABC) = m(arc AB) + m(arc BC)
= 125° + 110° = 235°
m (arc AC) = 360° - m (arc ACB)
= 360° - 235° = 125°
Similarly, m(arc ACB) = 360° - 125° = 235°
and m(arc BAC) = 360° - 110° = 250°
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Question 172 Marks
In the adjoining figure circle with centre D touches the sides of ∠ ACB at A and B. If ∠ ACB = 52°, find measure of ∠ ADB.
Answer
Solution is as follows:
Image
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Question 182 Marks
In the adjoining figure line PA is tangent at point A. Line PBC is a secant. If AP = 15 and BP = 10, find BC.
Image
Answer
12.3 units
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Question 192 Marks
In the adjoining figure, chord PQ and chord AB intersect at point M. If PM = AM, then prove that BM = QM.
Image
Answer
self
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Question 202 Marks
Tangents drawn at points A and C of a circle intersect each other in point P. If $\angle$APC = 50°, then find $\angle$ABC.
Image
Answer
65°
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Question 212 Marks
In the adjoining figure, O is the centre of the circle. Find the value of $\angle$ABP if $\angle$POB = 90°
Image
Answer
45°
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Question 232 Marks
In the adjoining figure, A circle with centre 'O' arc PQ = arc QR = arc PR. Find measure of each of above arcs.
Image
Answer
120°
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Question 242 Marks
The radii of two circles are 25 cm and 9 cm. The distance between their centres is 34 cm. Find the length of the common tangent segment to these circles.
Answer
30 units
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Question 252 Marks
In the adjoining figure circles with centres A and C touch internally at point T. Line AB is tangent to the smaller circle at point P. Point B lies on the bigger circle. Radii are 16 cm and 6 cm. Find AP.
Image
Answer
8 units
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Question 262 Marks
The circles which are not congruent touch externally. The sum of their areas is $130 \pi cm ^2$ and distance between their centres is 14 cm. Find the radii of the two circles.
Answer
11 cm, 3 cm
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Question 272 Marks
In the adjoining figure, O is the centre and seg AB is a diameter. At point C on the circle, the tangent CD is drawn. Line BD is tangent at B. Prove that seg OD || seg AC.
Image
Answer
self
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Question 282 Marks
Two tangents TP and TQ are drawn to a circle with centre $O$ from an external point T. Prove that $\angle PTQ =2 \angle OPQ$.
Answer
self
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Question 292 Marks
A circle with centre. O Point A is in the exterior of the circle. Line AP and line AQ are tangents at point P and point Q respectively P-A-S, Q-A-R $\angle$PAR = 130°. Find $\angle$AOP.
Image
Answer
65°
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Question 302 Marks
In the adjoining figure, point A is the centre of the circle AN = 10 cm. Line NM is tangent at M. MN = 5cm. Find the radius.
Image
Answer
$5 \sqrt{3}$
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Question 312 Marks
In the adjoining figure, point P is the centre of the circle and line AB is the tangent to the circle at T. The radius of the circle is 6 cm. Find PB if $\angle TPB =60^{\circ}$.
Image
Answer
12
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Question 402 Marks
In figure 3.89, line l touches the circle with centre O at point P. Q is the mid point of radius OP. RS is a chord through Q such that chords RS || line l. If RS = 12 find the radius of the circle.
Image
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Question 422 Marks
In figure 3.84, O is the centre of the circle. Seg AB, seg AC are tangent segments. Radius of the circle isr and l(AB) = r , Prove that, $\square$ ABOC is a square.
Image
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Question 432 Marks
In figure 3.76, seg PQ is a diameter of a circle with centre O. R is any point on the circle. seg RS⊥seg PQ.
Prove that, SR is the geometric mean of PS and SQ. [That is, SR² = PS ×SQ]
Image
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Question 442 Marks
Prove that, if two lines containing chords of a circle intersect each other outside the circle, then the measure of angle between them is half the difference in measures of the arcs intercepted by the angle.
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Question 462 Marks
A, B, C are any points on the circle with centre O.
(i) Write the names of all arcs formed due to these points.
(ii) If m arc (BC) = 110° and m arc (AB) = 125°, find measures of all remaining arcs.
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Question 472 Marks
In the adjoining figure circle with centre D touches the sides of ∠ ACB at A and B. If ∠ ACB = 52°, find measure of ∠ ADB.
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Question 482 Marks
In figure 3.52 , chords $\mathrm{PQ}$ and $\mathrm{RS}$ intersect at $\mathrm{T}$.
(i) Find $m$ (arc SQ) if $m \angle \mathrm{STQ}=58^{\circ}, m \angle \mathrm{PSR}=24^{\circ}$.
(ii) Verify,
$\angle \mathrm{STQ}=\frac{1}{2}[m(\operatorname{arc} \mathrm{PR})+m(\operatorname{arcSQ})]$
(iii) Prove that :
$\angle \mathrm{STQ}=\frac{1}{2}[m(\operatorname{arc} \mathrm{PR})+m(\operatorname{arcSQ})]$ for any measure of $\angle \mathrm{STQ}$.
(iv) Write in words the property in (iii).

Image

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Question 492 Marks
Two tangents TP and TQ are drawn to a circle with centre $O$ from an external point T. Prove that $\angle PTQ =2 \angle OPQ$.
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Question 502 Marks
The radii of two circles are 25 cm and 9 cm. The distance between their centres is 34 cm. Find the length of the common tangent segment to these circles.
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Question 512 Marks
The circles which are not congruent touch externally. The sum of their areas is $130 \pi cm ^2$ and distance between their centres is 14 cm. Find the radii of the two circles.
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Question 522 Marks
Tangents drawn at points A and C of a circle intersect each other in point P. If $\angle$APC = 50°, then find $\angle$ABC.
Image
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Question 532 Marks
In the adjoining figure, point P is the centre of the circle and line AB is the tangent to the circle at T. The radius of the circle is 6 cm. Find PB if $\angle TPB =60^{\circ}$.
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Question 542 Marks
In the adjoining figure, point A is the centre of the circle AN = 10 cm. Line NM is tangent at M. MN = 5cm. Find the radius.
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Question 552 Marks
In the adjoining figure, O is the centre of the circle. Find the value of $\angle$ABP if $\angle$POB = 90°
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Question 562 Marks
In the adjoining figure, O is the centre and seg AB is a diameter. At point C on the circle, the tangent CD is drawn. Line BD is tangent at B. Prove that seg OD || seg AC.
Image
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Question 572 Marks
In the adjoining figure line PA is tangent at point A. Line PBC is a secant. If AP = 15 and BP = 10, find BC.
Image
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Question 582 Marks
In the adjoining figure circles with centres A and C touch internally at point T. Line AB is tangent to the smaller circle at point P. Point B lies on the bigger circle. Radii are 16 cm and 6 cm. Find AP.
Image
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Question 592 Marks
In the adjoining figure, chord PQ and chord AB intersect at point M. If PM = AM, then prove that BM = QM.
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Question 602 Marks
In the adjoining figure, A circle with centre 'O' arc PQ = arc QR = arc PR. Find measure of each of above arcs.
Image
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Question 622 Marks
A circle with centre. O Point A is in the exterior of the circle. Line AP and line AQ are tangents at point P and point Q respectively P-A-S, Q-A-R $\angle$PAR = 130°. Find $\angle$AOP.
Image
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