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Class 10 Math Tangent to a Circle 2. The radius of a circle is 2.5 \mathrm{~cm} . \overline{A B} and \overline{C D} are two chords 3.9 \mathrm{~cm} apart. If m \overline{A B}=1.4 \mathrm{~cm} then measure the other chord.


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2. The radius of a circle is 2.5 \mathrm{~cm} . \overline{A B} and \overline{C D} are two chords 3.9 \mathrm{~cm} apart. If m \overline{A B}=1.4 \mathrm{~cm} then measure the other chord.

Example 1: \overline{A B} is a diameter of a given circle with centre O . Tangents are drawn at the end points A and B . Show that the two tangents are parallel.NOT FOR SALE - PESRP193Mathematics 10
Example 1:  \overline{A B}  is a diameter of a given circle with centre  O . Tangents are drawn at the end points  A  and  B . Show that the two tangents are parallel.NOT FOR SALE - PESRP193Mathematics 10

Example 1: \overline{A B} is a diameter of a given circle with centre O . Tangents are drawn at the end points A and B . Show that the two tangents are parallel.NOT FOR SALE - PESRP193Mathematics 10

2. The diameters of two concentric circles are 10 \mathrm{~cm} and 5 \mathrm{~cm} respectively. Look for the length of any chord of the outer circle which touches the inner one.(Hint) From the figure
2. The diameters of two concentric circles are  10 \mathrm{~cm}  and  5 \mathrm{~cm}  respectively. Look for the length of any chord of the outer circle which touches the inner one.(Hint) From the figure

2. The diameters of two concentric circles are 10 \mathrm{~cm} and 5 \mathrm{~cm} respectively. Look for the length of any chord of the outer circle which touches the inner one.(Hint) From the figure

2. The radius of a circle is 2.5 \mathrm{~cm} . \overline{A B} and \overline{C D} are two chords 3.9 \mathrm{~cm} apart. If m \overline{A B}=1.4 \mathrm{~cm} then measure the other chord.
2. The radius of a circle is  2.5 \mathrm{~cm} . \overline{A B}  and  \overline{C D}  are two chords  3.9 \mathrm{~cm}  apart. If  m \overline{A B}=1.4 \mathrm{~cm}  then measure the other chord.
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2. The radius of a circle is 2.5 \mathrm{~cm} . \overline{A B} and \overline{C D} are two chords 3.9 \mathrm{~cm} apart. If m \overline{A B}=1.4 \mathrm{~cm} then measure the other chord.

(iv) In the adjacent figure find half the perimeter of circle with centre O if \pi \simeq 3.1416 and m \overline{O A}=20 \mathrm{~cm} .(a) 31.42 \mathrm{~cm} (b) 62.832 \mathrm{~cm} (c) 125.65 \mathrm{~cm} (d) 188.50 \mathrm{~cm}
(iv) In the adjacent figure find half the perimeter of circle with centre  O  if  \pi \simeq 3.1416  and  m \overline{O A}=20 \mathrm{~cm} .(a)   31.42 \mathrm{~cm} (b)   62.832 \mathrm{~cm} (c)  125.65 \mathrm{~cm} (d)   188.50 \mathrm{~cm}

(iv) In the adjacent figure find half the perimeter of circle with centre O if \pi \simeq 3.1416 and m \overline{O A}=20 \mathrm{~cm} .(a) 31.42 \mathrm{~cm} (b) 62.832 \mathrm{~cm} (c) 125.65 \mathrm{~cm} (d) 188.50 \mathrm{~cm}

(x) Tangents drawn at the ends of diameter of a circle are ...... to each other.(a) parallel(b) non-parallel(c) collinear(d) perpendicular
(x) Tangents drawn at the ends of diameter of a circle are ...... to each other.(a) parallel(b) non-parallel(c) collinear(d) perpendicular

(x) Tangents drawn at the ends of diameter of a circle are ...... to each other.(a) parallel(b) non-parallel(c) collinear(d) perpendicular

1. \overline{A B} and \overline{C D} are two equal chords in a circle with centre O . H and K are respectively the mid points of the chords. Prove that \overline{H K} makes equal angles with \overline{A B} and \overline{C D} .
1.  \overline{A B}  and  \overline{C D}  are two equal chords in a circle with centre  O . H  and  K  are respectively the mid points of the chords. Prove that  \overline{H K}  makes equal angles with  \overline{A B}  and  \overline{C D} .

1. \overline{A B} and \overline{C D} are two equal chords in a circle with centre O . H and K are respectively the mid points of the chords. Prove that \overline{H K} makes equal angles with \overline{A B} and \overline{C D} .

(ix) A tangent line intersects the circle at:(a) three points(b) two points(c) single point(d) no point at all
(ix) A tangent line intersects the circle at:(a) three points(b) two points(c) single point(d) no point at all

(ix) A tangent line intersects the circle at:(a) three points(b) two points(c) single point(d) no point at all

3. The radii of two intersecting circles are 10 \mathrm{~cm} and 8 \mathrm{~cm} . If the length of their common chord is 6 \mathrm{~cm} then find the distance between the centres.
3. The radii of two intersecting circles are  10 \mathrm{~cm}  and  8 \mathrm{~cm} . If the length of their common chord is  6 \mathrm{~cm}  then find the distance between the centres.

3. The radii of two intersecting circles are 10 \mathrm{~cm} and 8 \mathrm{~cm} . If the length of their common chord is 6 \mathrm{~cm} then find the distance between the centres.

(viii) A circle has only one:(a) secant(b) chord(c) diameter(d) centre
(viii) A circle has only one:(a) secant(b) chord(c) diameter(d) centre

(viii) A circle has only one:(a) secant(b) chord(c) diameter(d) centre

(xi) The distance between the centres of two congruent touching circles externally is:(a) of zero length(b) the radius of each circle(c) the diameter of each circle(d) twice the diameter of each circle
(xi) The distance between the centres of two congruent touching circles externally is:(a) of zero length(b) the radius of each circle(c) the diameter of each circle(d) twice the diameter of each circle

(xi) The distance between the centres of two congruent touching circles externally is:(a) of zero length(b) the radius of each circle(c) the diameter of each circle(d) twice the diameter of each circle

1. Prove that the tangents drawn at the ends of a diameter in a given circle must be parallel.
1. Prove that the tangents drawn at the ends of a diameter in a given circle must be parallel.

1. Prove that the tangents drawn at the ends of a diameter in a given circle must be parallel.

(i) In the adjacent figure of the circle the line \overleftrightarrow{P T Q} is named as(a) an arc(b) a chord(c) a tangent(d) a secant
(i) In the adjacent figure of the circle the line  \overleftrightarrow{P T Q}  is named as(a) an arc(b) a chord(c) a tangent(d) a secant

(i) In the adjacent figure of the circle the line \overleftrightarrow{P T Q} is named as(a) an arc(b) a chord(c) a tangent(d) a secant

(xiii) In the adjoining figure there is a circle with centre O . If \overline{D C} / / diameter \overline{A B} and m \angle A O C=120^{\circ} then m \angle A C D is:(a) 40^{\circ} (b) 30^{\circ} (c) 50^{\circ}
(xiii) In the adjoining figure there is a circle with centre  O . If  \overline{D C} / /  diameter  \overline{A B}  and  m \angle A O C=120^{\circ}  then  m \angle A C D  is:(a)  40^{\circ} (b)  30^{\circ} (c)  50^{\circ}

(xiii) In the adjoining figure there is a circle with centre O . If \overline{D C} / / diameter \overline{A B} and m \angle A O C=120^{\circ} then m \angle A C D is:(a) 40^{\circ} (b) 30^{\circ} (c) 50^{\circ}

Example 2: In a circle the tangents drawn at the ends of a chord make equal angles with that chord.
Example 2: In a circle the tangents drawn at the ends of a chord make equal angles with that chord.

Example 2: In a circle the tangents drawn at the ends of a chord make equal angles with that chord.

(ii) In a circle with centre O if \overline{O T} is the radial segment and \overleftrightarrow{P T Q} is the tangent line then(a) \overline{O T} \perp \overleftrightarrow{P Q} (b) \overline{O T} \perp \overleftrightarrow{P Q} (c) \overline{O T} / / \overleftrightarrow{P Q} (d) \overline{O T} is right bisector of \overleftrightarrow{P Q}
(ii) In a circle with centre  O  if  \overline{O T}  is the radial segment and  \overleftrightarrow{P T Q}  is the tangent line then(a)   \overline{O T} \perp \overleftrightarrow{P Q} (b)  \overline{O T} \perp \overleftrightarrow{P Q} (c)  \overline{O T} / / \overleftrightarrow{P Q} (d)  \overline{O T}  is right bisector of  \overleftrightarrow{P Q}

(ii) In a circle with centre O if \overline{O T} is the radial segment and \overleftrightarrow{P T Q} is the tangent line then(a) \overline{O T} \perp \overleftrightarrow{P Q} (b) \overline{O T} \perp \overleftrightarrow{P Q} (c) \overline{O T} / / \overleftrightarrow{P Q} (d) \overline{O T} is right bisector of \overleftrightarrow{P Q}

3. \overleftrightarrow{A B} and \overleftrightarrow{C D} are the common tangents drawn to the pair of circles. If A and C are the points of tangency of 1st circle where B and D are the points of tangency of 2 nd circle then prove that \overline{A C} / / \overline{B D} .
3.   \overleftrightarrow{A B}  and  \overleftrightarrow{C D}  are the common tangents drawn to the pair of circles. If  A  and  C  are the points of tangency of 1st circle where  B  and  D  are the points of tangency of 2 nd circle then prove that  \overline{A C} / / \overline{B D} .

3. \overleftrightarrow{A B} and \overleftrightarrow{C D} are the common tangents drawn to the pair of circles. If A and C are the points of tangency of 1st circle where B and D are the points of tangency of 2 nd circle then prove that \overline{A C} / / \overline{B D} .

Example 1: Three circles touch in pairs externally. Prove that the perimeter of a triangle formed by joining centres is equal to the sum of their diameters.
Example 1: Three circles touch in pairs externally. Prove that the perimeter of a triangle formed by joining centres is equal to the sum of their diameters.

Example 1: Three circles touch in pairs externally. Prove that the perimeter of a triangle formed by joining centres is equal to the sum of their diameters.

(xii) In the adjacent circular figure with centre O and radius 5 \mathrm{~cm} the length of the chord intercepted at 4 \mathrm{~cm} away from the centre of this circle is:(a) 4 \mathrm{~cm} (b) 6 \mathrm{~cm} (c) 7 \mathrm{~cm}
(xii) In the adjacent circular figure with centre  O  and radius  5 \mathrm{~cm}  the length of the chord intercepted at  4 \mathrm{~cm}  away from the centre of this circle is:(a)  4 \mathrm{~cm} (b)  6 \mathrm{~cm} (c)  7 \mathrm{~cm}

(xii) In the adjacent circular figure with centre O and radius 5 \mathrm{~cm} the length of the chord intercepted at 4 \mathrm{~cm} away from the centre of this circle is:(a) 4 \mathrm{~cm} (b) 6 \mathrm{~cm} (c) 7 \mathrm{~cm}

(v) A line which has two points in common with a circle is called:(a) sine of a circle(b). cosine of a circle(c) tangent of a circle(d) secant of a circle
(v) A line which has two points in common with a circle is called:(a) sine of a circle(b). cosine of a circle(c) tangent of a circle(d) secant of a circle
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(v) A line which has two points in common with a circle is called:(a) sine of a circle(b). cosine of a circle(c) tangent of a circle(d) secant of a circle

4. Show that greatest chord in a circle is its diameter.
4. Show that greatest chord in a circle is its diameter.
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4. Show that greatest chord in a circle is its diameter.

(vi) A line which has only one point in common with a circle is called:(a) sine of a circle(b) cosine of a circle(c) tangent of a circle(d) secant of a circle
(vi) A line which has only one point in common with a circle is called:(a) sine of a circle(b) cosine of a circle(c) tangent of a circle(d) secant of a circle
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(vi) A line which has only one point in common with a circle is called:(a) sine of a circle(b) cosine of a circle(c) tangent of a circle(d) secant of a circle

(vii) Two tangents drawn to a circle from a point outside it are of(a) half(b) equal(c) double(d) triple
(vii) Two tangents drawn to a circle from a point outside it are of(a) half(b) equal(c) double(d) triple
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(vii) Two tangents drawn to a circle from a point outside it are of(a) half(b) equal(c) double(d) triple

(iii) In the adjacent figure find semicircular area if \pi \simeq 3.1416 and m \overline{O A}=20 \mathrm{~cm} .(a) 62.83 \mathrm{sq} \mathrm{cm} (b) 314.16 \mathrm{sq} \mathrm{cm} (c) 436.20 \mathrm{sq} \mathrm{cm} (d) 628.32 \mathrm{sq} \mathrm{cm}
(iii) In the adjacent figure find semicircular area if  \pi \simeq 3.1416  and  m \overline{O A}=20 \mathrm{~cm} .(a)  62.83 \mathrm{sq} \mathrm{cm} (b)   314.16 \mathrm{sq} \mathrm{cm} (c)   436.20 \mathrm{sq} \mathrm{cm} (d)  628.32 \mathrm{sq} \mathrm{cm}
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(iii) In the adjacent figure find semicircular area if \pi \simeq 3.1416 and m \overline{O A}=20 \mathrm{~cm} .(a) 62.83 \mathrm{sq} \mathrm{cm} (b) 314.16 \mathrm{sq} \mathrm{cm} (c) 436.20 \mathrm{sq} \mathrm{cm} (d) 628.32 \mathrm{sq} \mathrm{cm}

1. Two circles with radii 5 \mathrm{~cm} and 4 \mathrm{~cm} touch each other externally. Draw another circle with radius 2.5 \mathrm{~cm} touching the first pair externally.
1. Two circles with radii  5 \mathrm{~cm}  and  4 \mathrm{~cm}  touch each other externally. Draw another circle with radius  2.5 \mathrm{~cm}  touching the first pair externally.
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1. Two circles with radii 5 \mathrm{~cm} and 4 \mathrm{~cm} touch each other externally. Draw another circle with radius 2.5 \mathrm{~cm} touching the first pair externally.

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