Circles Archives - India Site

(a) If a, b, c are the sides of a right triangle where c is the hypotenuse, prove that the radius r of the circle which touches the sides of the triangle is given by r = /frac (a + b – c) – (2) (b) In the given figure, PB is a tangent to a circle with center O at B. AB is a chord of length 24 cm at a distance of 5 cm from the center. If the length of the tangent is 20 cm, find the length of OP.

CBSE, Circles, Class 10, Exercise 1, Maths, ML Aggarwal

Solution: (a) Let the circle touch the sides BC, CA and AB of the right triangle ABC at points D, E and F respectively, where BC = a, CA = b and AB = c (as showing in the given figure). As the...

The tangent to a circle of radius 6 cm from an external point P, is of length 8 cm. Calculate the distance of P from the nearest point of the circle.

CBSE, Circles, Class 10, Exercise 1, Maths, ML Aggarwal

Solution: Radius of the circle = 6 cm and length of tangent = 8 cm Let OP be the distance i.e. OA = 6 cm, AP = 8 cm . OA is the radius OA ⊥ AP Now In right ∆OAP, OP2 = OA2 + AP2 (By Pythagoras...

A point P is at a distance 13 cm from the center C of a circle and PT is a tangent to the given circle. If PT = 12 cm, find the radius of the circle.

CBSE, Circles, Class 10, Exercise 1, Maths, ML Aggarwal

Solution: CT is the radius CP = 13 cm and tangent PT = 12 cm CT is the radius and TP is the tangent CT is perpendicular TP Now in right angled triangle CPT, CP2 = CT2 + PT2 [using Pythagoras axiom]...

(a) In the figure given below, O is the center of the circle. If ∠BAD = 30°, find the values of p, q and r.

CBSE, Circles, Class 10, Exercise 1, Maths, maths, Maths, ML Aggarwal, ML Aggarwal

(a) In the figure given below, two circles intersect at points P and Q. If ∠A = 80° and ∠D = 84°, calculate (i) ∠QBC (ii) ∠BCP Solution: (i) ABCD is a cyclic quadrilateral ∠A + ∠C = 180o 30o + p =...

(a) In the figure (i) given below, O is the center of the circle. If ∠AOC = 150°, find (i) ∠ABC (ii) ∠ADC (b) In the figure (i) given below, AC is a diameter of the given circle and ∠BCD = 75°. Calculate the size of (i) ∠ABC (ii) ∠EAF.

CBSE, Circles, Class 10, Exercise 1, maths, Maths, ML Aggarwal

Solution: (a) Given, ∠AOC = 150° and AD = CD We know that an angle subtends by an arc of a circle at the center is twice the angle subtended by the same arc at any point on the remaining part of the...

(a) In the figure (i) given below, AB is a diameter of the circle APBR. APQ and RBQ are straight lines, ∠A = 35°, ∠Q = 25°. Find (i) ∠PRB (ii) ∠PBR (iii) ∠BPR. (b) In the figure (ii) given below, it is given that ∠ABC = 40° and AD is a diameter of the circle. Calculate ∠DAC.

CBSE, Circles, Class 10, Exercise 1, Maths, ML Aggarwal

Solution (a) (i) ∠PRB = ∠BAP (Angles in the same segment of the circle) ∴ ∠PRB = 35° (∵ ∠BAP = 35° given)

(a) In the figure (i) given below, M, A, B, N are points on a circle having centre O. AN and MB cut at Y. If ∠NYB = 50° and ∠YNB = 20°, find ∠MAN and the reflex angle MON. (b) In the figure (ii) given below, O is the centre of the circle. If ∠AOB = 140° and ∠OAC = 50°, find (i) ∠ACB (ii) ∠OBC (iii) ∠OAB (iv) ∠CBA

CBSE, Circles, Class 10, Exercise 1, Maths, ML Aggarwal

Solution (a) ∠NYB = 50°, ∠YNB = 20°. In ∆YNB, ∠NYB + ∠YNB + ∠YBN = 180o 50o + 20o + ∠YBN = 180o ∠YBN + 70o = 180o ∠YBN = 180o – 70o = 110o But ∠MAN = ∠YBN (Angles in the same segment) ∠MAN = 110o...

In the figure (i) given below, calculate the values of x and y. (b) In the figure (ii) given below, O is the centre of the circle. Calculate the values of x and y.

CBSE, Circles, Class 10, Exercise 1, Maths, ML Aggarwal

(a) ABCD is cyclic Quadrilateral ∠B + ∠D = 1800 Y + 400 + 45o = 180o (y + 85o = 180o) Y = 180o – 85o = 95o ∠ACB = ∠ADB xo = 40 (a) Arc ADC Subtends ∠AOC at the centre and ∠ ABC at the remaining part...

Two congruent circles have their centers at O and P respectively . Line segment OP has the midpoint M. A straight line is drawn through M cutting the two circles at the points A, B, C and D. We need to Prove that the chords of the circle AB and CD are equal.

Circles, Class 10, Frank, Guides, ICSE, Maths

From the question it is given that, O and P are the centers of the congruent circles. Line segment OP has the midpoint M To Prove:, Chord AB and CD are equal. Then, draw $OQ\bot AB$ and $PR\bot CD$....

We have to prove that the line segment joining the midpoints of two parallel chords of a circle passes through its center.

Circles, Class 10, Frank, Guides, ICSE, Maths

Let’s assume AB and CD is the two parallel chords of the circle having Q and P as their mid-points, respectively. Let the Circle has the center O. Construction: Join OP and OQ and draw...

Chords AB and CD has midpoints M and N respectively. The line MN passes through the center O. We have to Prove that $AB||CD$.

Circles, Class 10, Frank, ICSE, Maths

From the figure given it is clear, $AM=MB$ $CN=ND$ Therefore, it is clear $OM\bot AB$ Then, we can say $ON\bot CD$ So we know, a line bisecting the chord and passing through the centre of the circle...

Two equal chords are PQ and QR and the diameter of the circle is drawn through Q. We need to Prove that the diameter bisect $\angle PQR$.

Circles, Class 10, Frank, Guides, ICSE, Maths

Assume that QT be the diameter of $\angle PQR$ Given that $PQ=QR$ So, $OM=ON$ Now, consider the $\vartriangle OMQ$and $\vartriangle ONQ$, $\angle OMQ=\angle ONQ$ …[both angles are equal...

In fig, the center of the circle is O. PQ and RS are the two equal chords of the circle which, when produced, meet at T outside the circle. So we need to Prove that (a) TP = TR, (B) TQ = TS

Circles, Class 10, Frank, Guides, ICSE, Maths

ATQ, $PQ=RS$ To Prove: $TP=TR$ and $TQ=TS$ Then, Draw $OA\bot PQ$and $OB\bot RS$, Since equal chords are equidistant from the circle therefore we can say that, $PQ=RS$ ⇒ (chords are equal as it is...

In fig, AB a chord of the circle is of length $18cm$. It is perpendicularly bisected at M by PQ. If $MQ=3cm$, find the length of PQ(it is diameter of circle).

Circles, Class 10, Frank, ICSE, Maths

ATQ, Length of chord $AB=18cm$ $MQ=3cm$ From the figure, $AM=MB=5cm$ATQ perpendicular from center to a chord bisects the chord. Let, $OA=OQ=p$ $OM=(r–3)$ Consider the $\vartriangle OMA$, Apply...

AB and CD are two equal chords of a circle which is intersecting at P as shown in fig. P is joined to O, the center of the circle. So we need to Prove that OP bisects $\angle CPB$

Circles, Class 10, Frank, Guides, ICSE, Maths

Construction: In the given figure draw perpendiculars OM and ON to AB and CD respectively. Now, we have to consider the $\vartriangle OMP$ and $\vartriangle ONP$, OP = OP … [common side for both...

Two chords of lengths $10cm$ and $24cm$ are drawn parallel to each other in a circle. If they are on the same side of the center and the distance between them is $17cm$, so need to find the radius of the circle.

Circles, Class 10, Frank, Guides, ICSE, Maths

From the figure, $AM=MB=5cm$ and $CN=ND=12cm$ATQ, perpendicular from center to a chord bisects the chord. Let, OA = OC = p OM = y $ON=17– y$ Consider the $\vartriangle CNO$, Apply Pythagoras...

Two chords AB and CD of lengths $6cm$ and $12cm$ are drawn parallel inside the circle. If the distance between the chords of the circle is $3cm$,so we need to find the radius of the circle.

Circles, Class 10, Frank, Guides, ICSE, Maths

From figure, $AM=MB=3cm$ and $CN=ND=6cm$ATQ, perpendicular from center to a chord bisects the chord. Let, $OA=OC=p$ $OM=y$ $ON=3–y$ Consider the $\vartriangle CNO$, Apply Pythagoras theorem,...

Two circles of radii $5cm$ and $3cm$ with centers O and P touch each other internally. If the perpendicular bisector of the line segment OP meets the circumference of the larger circle at A and B, find the length of AB (chord of the larger circle).

Circles, Class 10, Frank, Guides, ICSE, Maths

From the question it is given that, Radius of bigger circle $=5cm$ Radius of smaller circle $=3cm$ Then, $OA=AH=5cm$… [both are radius of bigger circle] $PH=3cm$ … [radius of smaller circle]...

A chord of length $8cm$ is drawn inside a circle of radius $46cm$.Find the perpendicular distance of the chord from the center of the circle having diameter $12cm$.

Circles, Class 10, Frank, ICSE, Maths

ATQ,, Length of chord is $8cm$. $Radius=6cm$. From the figure we can say that, $PR=RQ=4cm$ATQ, perpendicular from center to a chord bisects the chord. Consider the ΔPRO, Apply Pythagoras theorem,...

A chord of length $6cm$ is at a distance of $7.2cm$ from the center of a circle. Another chord of the same circle is of length $14.4cm$. Find the chord’s distance from the center of the circle.

Circles, Class 10, Frank, Guides, ICSE, Maths

We can say, $PM=MQ=3cm$ATQ, perpendicular from center to a chord bisects the chord. Consider the ΔPMO, Apply Pythagoras theorem, \[O{{P}^{2}}=P{{M}^{2}}+O{{M}^{2}}\] $O{{P}^{2}}={{\left( 3...

A chord of length $16.8cm$ is at a distance of $11.2cm$ from the center of a circle. Find the length of the chord of the same circle which is at a distance of $8.4cm$ from the center of the circle.

Circles, Class 10, Frank, ICSE, Maths

From the figure we can say that, $PM=MQ=8.4cm$We know according to question perpendicular from center to a chord bisects the chord. Consider the $\vartriangle PMO$, Apply Pythagoras theorem,...

Find the diameter of the circle if the length of a chord is $3.2cm$ and the distance of the chord from the center is $1.2cm$.

Circles, Class 10, Frank, Guides, ICSE, Maths

ATQ, length of a chord is $3.2cm$ Distance of the circle from the center is $1.2cm$ Then, So , $PR=RQ=1.6cm$We know that perpendicular from center to a chord bisects the chord. Consider the...

What will be the length of the chord of a circle in each of the following when:(iii) The circle has the Radius is $6.5cm$ and the distance of the circle from the center is $2.5cm$.

Circles, Class 10, Frank, Guides, ICSE, Maths

Given : Radius $=6.5cm$ Distance from the center is $2.5cm$ So, $PR=RQ$We know that perpendicular from center to a chord bisects the chord. Consider the $\vartriangle PRQ$, Apply Pythagoras...

What will be the length of the chord of a circle in each of the following when:(i) The circle has the radius $13cm$ and the distance of the chord from the center is $12cm$(ii) The circle of the radius is $1.7cm$ and the distance of the chord from the center is $1.5cm$.

Circles, Class 10, Frank, Guides, ICSE, Maths

Given: Radius $=13cm$ Distance of chord from the center is $12cm$ Therefore, $PR=RQ$We know that perpendicular from center to a chord bisects the chord of the circle. Consider the...

The tangent to a circle of radius 6 cm from an external point P, is of length 8 cm. Calculate the distance of P from the nearest point of the circle.

A point P is at a distance 13 cm from the center C of a circle and PT is a tangent to the given circle. If PT = 12 cm, find the radius of the circle.

(a) In the figure given below, O is the center of the circle. If ∠BAD = 30°, find the values of p, q and r.

(a) In the figure (i) given below, O is the center of the circle. If ∠AOC = 150°, find (i) ∠ABC (ii) ∠ADC (b) In the figure (i) given below, AC is a diameter of the given circle and ∠BCD = 75°. Calculate the size of (i) ∠ABC (ii) ∠EAF.

(a) In the figure (i) given below, AB is a diameter of the circle APBR. APQ and RBQ are straight lines, ∠A = 35°, ∠Q = 25°. Find (i) ∠PRB (ii) ∠PBR (iii) ∠BPR. (b) In the figure (ii) given below, it is given that ∠ABC = 40° and AD is a diameter of the circle. Calculate ∠DAC.

In the figure (i) given below, calculate the values of x and y. (b) In the figure (ii) given below, O is the centre of the circle. Calculate the values of x and y.

Two congruent circles have their centers at O and P respectively . Line segment OP has the midpoint M. A straight line is drawn through M cutting the two circles at the points A, B, C and D. We need to Prove that the chords of the circle AB and CD are equal.

We have to prove that the line segment joining the midpoints of two parallel chords of a circle passes through its center.

Chords AB and CD has midpoints M and N respectively. The line MN passes through the center O. We have to Prove that $AB||CD$.

Two equal chords are PQ and QR and the diameter of the circle is drawn through Q. We need to Prove that the diameter bisect $\angle PQR$.

In fig, the center of the circle is O. PQ and RS are the two equal chords of the circle which, when produced, meet at T outside the circle. So we need to Prove that (a) TP = TR, (B) TQ = TS

In fig, AB a chord of the circle is of length $18cm$. It is perpendicularly bisected at M by PQ. If $MQ=3cm$, find the length of PQ(it is diameter of circle).

AB and CD are two equal chords of a circle which is intersecting at P as shown in fig. P is joined to O, the center of the circle. So we need to Prove that OP bisects $\angle CPB$

Two chords of lengths $10cm$ and $24cm$ are drawn parallel to each other in a circle. If they are on the same side of the center and the distance between them is $17cm$, so need to find the radius of the circle.

Two chords AB and CD of lengths $6cm$ and $12cm$ are drawn parallel inside the circle. If the distance between the chords of the circle is $3cm$,so we need to find the radius of the circle.

Two circles of radii $5cm$ and $3cm$ with centers O and P touch each other internally. If the perpendicular bisector of the line segment OP meets the circumference of the larger circle at A and B, find the length of AB (chord of the larger circle).

A chord of length $8cm$ is drawn inside a circle of radius $46cm$.Find the perpendicular distance of the chord from the center of the circle having diameter $12cm$.

A chord of length $6cm$ is at a distance of $7.2cm$ from the center of a circle. Another chord of the same circle is of length $14.4cm$. Find the chord’s distance from the center of the circle.

A chord of length $16.8cm$ is at a distance of $11.2cm$ from the center of a circle. Find the length of the chord of the same circle which is at a distance of $8.4cm$ from the center of the circle.

Find the diameter of the circle if the length of a chord is $3.2cm$ and the distance of the chord from the center is $1.2cm$.

What will be the length of the chord of a circle in each of the following when:(iii) The circle has the Radius is $6.5cm$ and the distance of the circle from the center is $2.5cm$.

What will be the length of the chord of a circle in each of the following when:(i) The circle has the radius $13cm$ and the distance of the chord from the center is $12cm$(ii) The circle of the radius is $1.7cm$ and the distance of the chord from the center is $1.5cm$.

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