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Common Tangents

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**Two circles which intersect at two points**

Common Tangents Two circles which intersect at two points Common tangent: AB and CD A B P Q Properties: Parallel to PQ Same length that is AB = CD C D

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**Two circles which intersect at two points**

Common Tangents Two circles which intersect at two points A B C D F P Q Common tangent: AB and CD

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**Two cirlces which intersect at two points**

Common Tangents Two cirlces which intersect at two points A B F P Q D Properties: Intersect at point F AB = CD C

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**Two circles which intersect at only one point**

Common Tangents Two circles which intersect at only one point A B F Common tangent: AB Q P Properties: Perpendicular to FQP

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**Two circles which intersect at only one point**

Common Tangents Two circles which intersect at only one point Common tangent: AB and CD A B P Q Properties: Parallel to PQ AB = CD C D

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**Two circles which intersect at only one point**

Common Tangents Two circles which intersect at only one point Common tangent: EF A E B P Q Properties: Perpendicular to PQ C F D

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**Two circles which intersect at only one point**

Common Tangents Two circles which intersect at only one point A Common tangent: AB and CD B G P Q D Properties: Intersect at point G AB = CD C

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**Two circles which intersect at only one point**

Common Tangents Two circles which intersect at only one point A Common tangent: EF E B G P Q D F Properties: Perpendicular to PQ C

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**Two circles which do not intersect each other**

Common Tangents Two circles which do not intersect each other Common tangent: AB and CD A B P Q Properties: Parallel to PQ AB = CD C D

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**Two circles which do not intersect each other**

Common Tangents Two circles which do not intersect each other Common tangent: EH and FG A B E G P Q F H Properties: Intersect at line PQ EH = FG C D

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**Two circles which do not intersect each other**

Common Tangents Two circles which do not intersect each other A B Common tangent: AB and CD F P Q C D Properties: Intersect at point F Same length that is AB = CD

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**Two circles which do not intersect each other**

Common Tangents Two circles which do not intersect each other A B Common tangent: GH and JK H G K J F P Q C D Properties: Intersect at the line PQ Same length that is GH = JK

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**Common Tangents Solving problems**

In the diagram, P and Q are the centres of two circles with radii 9 cm and 4 cm respectively. MN is a common tangent to the circles. Calculate Q M N H P x (a) the length of MN, (b) the value of x, (c) the perimeter of the shaded region. (Assume = 3.142)

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**Common Tangents Solving problems P Q T Solution: (a) PQ = 9 + 4**

= 13 cm PT = 9 – 4 = 5 cm P H In ∆PQT, TQ2 = PQ2 – PT2 x Q T = 132 – 52 TQ = 12 cm M N ஃ MN = 12 cm

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**Common Tangents Solving problems P Q T Solution: (b) tan x = = = 2.4**

H x Q T M N

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**Common Tangents Solving problems P Q T Solution: (c) Length of arc HM**

= × 2 × × 9 = cm HQN = 180° – 67.4° P = 112.6° H x Length of arc HN = Q T × 2 × × 4 = 7.86 cm M N Perimeter of the shaded region = = cm

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The End

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