Page 1 (www.tiwariacademy.net) (Class – X) A Free web support in Education 1 Exercise 11.2 Question 1: Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths. Give the justification of the construction. Answer 1: A pair of tangents to the given circle can be constructed as follows. Step 1 Taking any point O of the given plane as centre, draw a circle of 6 cm radius. Locate a point P, 10 cm away from O. Join OP. Step 2 Bisect OP. Let M be the mid-point of PO. Step 3 Taking M as centre and MO as radius, draw a circle. Step 4 Let this circle intersect the previous circle at point Q and R. Step 5 Join PQ and PR. PQ and PR are the required tangents. The lengths of tangents PQ and PR are 8 cm each. Justification The construction can be justified by proving that PQ and PR are the tangents to the circle (whose centre is O and radius is 6 cm). For this, join OQ and OR. Page 2 (www.tiwariacademy.net) (Class – X) A Free web support in Education 1 Exercise 11.2 Question 1: Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths. Give the justification of the construction. Answer 1: A pair of tangents to the given circle can be constructed as follows. Step 1 Taking any point O of the given plane as centre, draw a circle of 6 cm radius. Locate a point P, 10 cm away from O. Join OP. Step 2 Bisect OP. Let M be the mid-point of PO. Step 3 Taking M as centre and MO as radius, draw a circle. Step 4 Let this circle intersect the previous circle at point Q and R. Step 5 Join PQ and PR. PQ and PR are the required tangents. The lengths of tangents PQ and PR are 8 cm each. Justification The construction can be justified by proving that PQ and PR are the tangents to the circle (whose centre is O and radius is 6 cm). For this, join OQ and OR. (www.tiwariacademy.net) (Class – X) A Free web support in Education 2 ?PQO is an angle in the semi-circle. We know that angle in a semi-circle is a right angle. ? ?PQO = 90° ? OQ ? PQ Since OQ is the radius of the circle, PQ has to be a tangent of the circle. Similarly, PR is a tangent of the circle Question 2: Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation. Give the justification of the construction. Answer 2: Tangents on the given circle can be drawn as follows. Step 1 Draw a circle of 4 cm radius with centre as O on the given plane. Step 2 Draw a circle of 6 cm radius taking O as its centre. Locate a point P on this circle and join OP. Step 3 Bisect OP. Let M be the mid-point of PO. Step 4 Taking M as its centre and MO as its radius, draw a circle. Let it intersect the given circle at the points Q and R. Page 3 (www.tiwariacademy.net) (Class – X) A Free web support in Education 1 Exercise 11.2 Question 1: Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths. Give the justification of the construction. Answer 1: A pair of tangents to the given circle can be constructed as follows. Step 1 Taking any point O of the given plane as centre, draw a circle of 6 cm radius. Locate a point P, 10 cm away from O. Join OP. Step 2 Bisect OP. Let M be the mid-point of PO. Step 3 Taking M as centre and MO as radius, draw a circle. Step 4 Let this circle intersect the previous circle at point Q and R. Step 5 Join PQ and PR. PQ and PR are the required tangents. The lengths of tangents PQ and PR are 8 cm each. Justification The construction can be justified by proving that PQ and PR are the tangents to the circle (whose centre is O and radius is 6 cm). For this, join OQ and OR. (www.tiwariacademy.net) (Class – X) A Free web support in Education 2 ?PQO is an angle in the semi-circle. We know that angle in a semi-circle is a right angle. ? ?PQO = 90° ? OQ ? PQ Since OQ is the radius of the circle, PQ has to be a tangent of the circle. Similarly, PR is a tangent of the circle Question 2: Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation. Give the justification of the construction. Answer 2: Tangents on the given circle can be drawn as follows. Step 1 Draw a circle of 4 cm radius with centre as O on the given plane. Step 2 Draw a circle of 6 cm radius taking O as its centre. Locate a point P on this circle and join OP. Step 3 Bisect OP. Let M be the mid-point of PO. Step 4 Taking M as its centre and MO as its radius, draw a circle. Let it intersect the given circle at the points Q and R. (www.tiwariacademy.net) (Class – X) A Free web support in Education 3 Step 5 Join PQ and PR. PQ and PR are the required tangents. It can be observed that PQ and PR are of length 4.47 cm each. In ?PQO, Since PQ is a tangent, ?PQO = 90° PO = 6 cm QO = 4 cm Applying Pythagoras theorem in ?PQO, we obtain PQ 2 + QO 2 = PQ 2 PQ 2 + (4) 2 = (6) 2 PQ 2 + 16 = 36 PQ 2 = 36 - 16 PQ 2 = 20 PQ PQ = 4.47 cm Justification The construction can be justified by proving that PQ and PR are the tangents to the circle (whose centre is O and radius is 4 cm). For this, let us join OQ and OR. Page 4 (www.tiwariacademy.net) (Class – X) A Free web support in Education 1 Exercise 11.2 Question 1: Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths. Give the justification of the construction. Answer 1: A pair of tangents to the given circle can be constructed as follows. Step 1 Taking any point O of the given plane as centre, draw a circle of 6 cm radius. Locate a point P, 10 cm away from O. Join OP. Step 2 Bisect OP. Let M be the mid-point of PO. Step 3 Taking M as centre and MO as radius, draw a circle. Step 4 Let this circle intersect the previous circle at point Q and R. Step 5 Join PQ and PR. PQ and PR are the required tangents. The lengths of tangents PQ and PR are 8 cm each. Justification The construction can be justified by proving that PQ and PR are the tangents to the circle (whose centre is O and radius is 6 cm). For this, join OQ and OR. (www.tiwariacademy.net) (Class – X) A Free web support in Education 2 ?PQO is an angle in the semi-circle. We know that angle in a semi-circle is a right angle. ? ?PQO = 90° ? OQ ? PQ Since OQ is the radius of the circle, PQ has to be a tangent of the circle. Similarly, PR is a tangent of the circle Question 2: Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation. Give the justification of the construction. Answer 2: Tangents on the given circle can be drawn as follows. Step 1 Draw a circle of 4 cm radius with centre as O on the given plane. Step 2 Draw a circle of 6 cm radius taking O as its centre. Locate a point P on this circle and join OP. Step 3 Bisect OP. Let M be the mid-point of PO. Step 4 Taking M as its centre and MO as its radius, draw a circle. Let it intersect the given circle at the points Q and R. (www.tiwariacademy.net) (Class – X) A Free web support in Education 3 Step 5 Join PQ and PR. PQ and PR are the required tangents. It can be observed that PQ and PR are of length 4.47 cm each. In ?PQO, Since PQ is a tangent, ?PQO = 90° PO = 6 cm QO = 4 cm Applying Pythagoras theorem in ?PQO, we obtain PQ 2 + QO 2 = PQ 2 PQ 2 + (4) 2 = (6) 2 PQ 2 + 16 = 36 PQ 2 = 36 - 16 PQ 2 = 20 PQ PQ = 4.47 cm Justification The construction can be justified by proving that PQ and PR are the tangents to the circle (whose centre is O and radius is 4 cm). For this, let us join OQ and OR. (www.tiwariacademy.net) (Class – X) A Free web support in Education 4 ?PQO is an angle in the semi-circle. We know that angle in a semi-circle is a right angle. ? ?PQO = 90° ? OQ ? PQ Since OQ is the radius of the circle, PQ has to be a tangent of the circle. Similarly, PR is a tangent of the circle Question 3: Draw a circle of radius 3 cm. Take two points P and Q on one of its extended diameter each at a distance of 7 cm from its centre. Draw tangents to the circle from these two points P and Q. Give the justification of the construction. Answer 3: The tangent can be constructed on the given circle as follows. Step 1 Taking any point O on the given plane as centre, draw a circle of 3 cm radius. Step 2 Take one of its diameters, PQ, and extend it on both sides. Locate two points on this diameter such that OR = OS = 7 cm Step 3 Bisect OR and OS. Let T and U be the mid-points of OR and OS respectively. Step 4 Taking T and U as its centre and with TO and UO as radius, draw two circles. These two circles will intersect the circle at point V, W, X, Y respectively. Join RV, RW, SX, and SY. These are the required tangents. Page 5 (www.tiwariacademy.net) (Class – X) A Free web support in Education 1 Exercise 11.2 Question 1: Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths. Give the justification of the construction. Answer 1: A pair of tangents to the given circle can be constructed as follows. Step 1 Taking any point O of the given plane as centre, draw a circle of 6 cm radius. Locate a point P, 10 cm away from O. Join OP. Step 2 Bisect OP. Let M be the mid-point of PO. Step 3 Taking M as centre and MO as radius, draw a circle. Step 4 Let this circle intersect the previous circle at point Q and R. Step 5 Join PQ and PR. PQ and PR are the required tangents. The lengths of tangents PQ and PR are 8 cm each. Justification The construction can be justified by proving that PQ and PR are the tangents to the circle (whose centre is O and radius is 6 cm). For this, join OQ and OR. (www.tiwariacademy.net) (Class – X) A Free web support in Education 2 ?PQO is an angle in the semi-circle. We know that angle in a semi-circle is a right angle. ? ?PQO = 90° ? OQ ? PQ Since OQ is the radius of the circle, PQ has to be a tangent of the circle. Similarly, PR is a tangent of the circle Question 2: Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation. Give the justification of the construction. Answer 2: Tangents on the given circle can be drawn as follows. Step 1 Draw a circle of 4 cm radius with centre as O on the given plane. Step 2 Draw a circle of 6 cm radius taking O as its centre. Locate a point P on this circle and join OP. Step 3 Bisect OP. Let M be the mid-point of PO. Step 4 Taking M as its centre and MO as its radius, draw a circle. Let it intersect the given circle at the points Q and R. (www.tiwariacademy.net) (Class – X) A Free web support in Education 3 Step 5 Join PQ and PR. PQ and PR are the required tangents. It can be observed that PQ and PR are of length 4.47 cm each. In ?PQO, Since PQ is a tangent, ?PQO = 90° PO = 6 cm QO = 4 cm Applying Pythagoras theorem in ?PQO, we obtain PQ 2 + QO 2 = PQ 2 PQ 2 + (4) 2 = (6) 2 PQ 2 + 16 = 36 PQ 2 = 36 - 16 PQ 2 = 20 PQ PQ = 4.47 cm Justification The construction can be justified by proving that PQ and PR are the tangents to the circle (whose centre is O and radius is 4 cm). For this, let us join OQ and OR. (www.tiwariacademy.net) (Class – X) A Free web support in Education 4 ?PQO is an angle in the semi-circle. We know that angle in a semi-circle is a right angle. ? ?PQO = 90° ? OQ ? PQ Since OQ is the radius of the circle, PQ has to be a tangent of the circle. Similarly, PR is a tangent of the circle Question 3: Draw a circle of radius 3 cm. Take two points P and Q on one of its extended diameter each at a distance of 7 cm from its centre. Draw tangents to the circle from these two points P and Q. Give the justification of the construction. Answer 3: The tangent can be constructed on the given circle as follows. Step 1 Taking any point O on the given plane as centre, draw a circle of 3 cm radius. Step 2 Take one of its diameters, PQ, and extend it on both sides. Locate two points on this diameter such that OR = OS = 7 cm Step 3 Bisect OR and OS. Let T and U be the mid-points of OR and OS respectively. Step 4 Taking T and U as its centre and with TO and UO as radius, draw two circles. These two circles will intersect the circle at point V, W, X, Y respectively. Join RV, RW, SX, and SY. These are the required tangents. (www.tiwariacademy.net) (Class – X) A Free web support in Education 5 Justification The construction can be justified by proving that RV, RW, SY, and SX are the tangents to the circle (whose centre is O and radius is 3 cm). For this, join OV, OW, OX, and OY. ?RVO is an angle in the semi-circle. We know that angle in a semi-circle is a right angle. ? ?RVO = 90° ? OV ? RV Since OV is the radius of the circle, RV has to be a tangent of the circle. Similarly, OW, OX, and OY are the tangents of the circleRead More

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