Soal Matematika : CMC - Circle Geometry

1. In a circle with centre O, two chords AC and BD intersect at P. Show that ∠APB = ½ (∠AOB + ∠COD).

2. If the points A, B, C and D are any 4 points on a circle and P, Q, R and S are the midpoints of the arcs AB, BC, CD and DA respectively, show that PR is perpendicular to QS.

3. Calculate the value of x.

4. The three vertices of triangle ABC lie on a circle. Chords AX, BY, CZ are drawn within the interior angles A, B, C of the triangle. Show that the chords AX, BY and CZ are the altitudes of triangle XY Z if and only if they are the angle bisectors of triangle ABC.

5. As shown in the diagram, a circle with centre A and radius 9 is tangent to a smaller circle with centre D and radius 4. Common tangents EF and BC are drawn to the circles making points of contact at E, B and C.

Determine the length of EF.

6. In the diagram, two circles are tangent at A and have a common tangent touching them at B and C respectively.

(a) Show that ∠BAC = 90⁰. (Hint: with touching circles it is usual to draw the common tangent at the point of contact!)

(b) If BA is extended to meet the second circle at D show that CD is a diameter.

7. If ABCD is a quadrilateral with an inscribed circle as shown, prove that AB + CD = AD + BC.

8. In this diagram, the two circles are tangent at A. The line BDC is tangent to the smaller circle. Show that AD bisects ∠BAC.

9. Starting at point A₁ on a circle, a particle moves to A₂ on the circle along chord A₁A₂ which makes a clockwise angle of 35⁰ to the tangent to the circle at A₁. From A₂ the particle moves to A₃ along chord A₂A₃ which makes a clockwise angle of 37⁰ to the tangent at A₂. The particle continues in this way. From A_k it moves to A_k+1 along chord A_kA_k+1 which makes a clockwise angle of (33 + 2k)⁰ to the tangent to the circle at A_k. After several trips around the circle, the particle returns to A₁ for the first time along chord A_nA₁. Find the value of n.