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Pythagoras & Circles. Problem Solving.

B. O. 8.7cm. 16.4cm. K. A. O. r. M. A. r- 1. B. 8. L. Pythagoras & Circles. Problem Solving. O. 10cm. M. A. 4cm. B. O. 10cm. 6cm. L. M. x. B. Chord and Radius Problem.

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Pythagoras & Circles. Problem Solving.

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  1. B O 8.7cm 16.4cm K A O r M A r- 1 B 8 L Pythagoras & Circles.Problem Solving.

  2. O 10cm M A 4cm B O 10cm 6cm L M x B Chord and Radius Problem. The radius OL bisects the chord AB. If the radius of the circle is 10cm and ML = 4cm as shown, then calculate the length of the chord AB. Solution. A radius which bisects a chord , meets the chord at 90o. Hence triangle OMB is right angled. OM = 10 – 4 = 6cm Now use the theorem of Pythagoras to solve the problem.

  3. O 10cm 6cm M x B x2 + 62 = 102 The length of MB is 8cm which gives us the chord length of 16cm as required. x2 + 36 = 100 x2 = 100 – 36 x = 64 x = 8

  4. O r r-8 r-1 B M O r M A r- 1 B 8 L Chord and Radius Problem 2. Solution. Consider the triangle OMB. Find the radius of this circle, given that the radius OL bisects the chord AB as shown and ML = 8cm. NB: OM = r - 8 Now use the theorem of Pythagoras to solve the problem.

  5. O r r-8 r-1 B M (r-8)2 + (r-1) 2 = r 2 r 2 – 16r+ 64 + r 2 – 2r + 1 = r 2 r 2 – 18r + 65= 0 r = 5 should be rejected as it creates a side of –3cm , which clearly cannot be the case. Hence the solution is radius = 13cm as required. ( r – 13 ) ( r - 5 ) = 0 r = 13 r = 5

  6. B O 8.7cm 16.4cm O K X 8.7cm A K B 16.4cm Tangents To Circle Problem. Solution. The tangent AB meets the radius OK at 90o on the circumference. A circle centre O of radius 8.7cm meets a tangent AB at the point K. If KB = 16.4cm, then calculate the length of OB. By Pythagoras: X2 = 8.72 + 16.42 X = 344.65 = 18.6 cm OB = 18.6cm as required.

  7. 12cm C B 18cm 12cm A C B X 18cm A Circle Diameter Problem. Solution. All triangles using a diameter as one side whose third vertex touches the circumference are right angled on the circumference AB= 18cm is the diameter of a circle. The triangle ABC meets the circumference at C. Calculate the length of side CA. By Pythagoras: X2 + 122 =182 X = 13.4cm CA is 13.4cm as required.

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