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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 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.

(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

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

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

(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.

(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}

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}

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.

(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

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

### 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

(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

(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}

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.