A Math Plane Geometry Practices

Page 1 Paradigm Specialising in O Level Mathematics Maths Secrets Plane Geometry 1 In the figure, 𝑋𝑌𝑍 is a straight line that is tangent to the circle at 𝑋. 𝑋𝑄 bisects ∠𝑅𝑋𝑍 and cuts the circle at 𝑆. 𝑅𝑆 produced meets 𝑋𝑍 at 𝑌 and 𝑍𝑅 = 𝑋𝑅. Prove that a) 𝑆𝑅 = 𝑆𝑋, b) a circle can be drawn passing through 𝑍, 𝑌, 𝑆 and 𝑄. 2 In the diagram, 𝐴, 𝐵, 𝐶 and 𝐷 are points on the circle centre 𝑂. 𝐴𝑃 and 𝐵𝑃 are tangents to the circle at 𝐴 and 𝐵 respectively. 𝐷𝑄 and 𝐶𝑄 are tangents to the circle at 𝐷 and 𝐶 respectively. 𝑃𝑂𝑄 is a straight line. (i)Prove that angle 𝐶𝑃𝐷 = 2 × angle 𝐶𝐷𝑄. (ii)Make a similar deduction about angle 𝐴𝑂𝐵. (iii)Prove that 2 × angle 𝑂𝐴𝐷 = angle 𝐶𝐷𝑄 + angle 𝐵𝐴𝑃 3 The diagram shows two intersecting circles, 𝐶1 and 𝐶2. 𝐶1 passes through the vertices of the triangle 𝐴𝐵𝐷. The tangents to 𝐶1 at 𝐴 and 𝐵 intersect at the point 𝑄 on 𝐶2. A line os drawn from 𝑄 to intersect the line 𝐴𝐷 at 𝐸 on 𝐶2. Prove that (i)𝑄𝐸 bisects angle 𝐴𝐸𝐵, (ii)𝐸𝐵 = 𝐸𝐷, (iii)𝐵𝐷 is parallel to 𝑄𝐸. 4 In the diagram, 𝐴, 𝐵 and 𝐶 are three points on the circle such that 𝐴𝐵 is the diameter of the circle and 𝑊 is the midpoint of 𝐴𝐶. 𝐴𝐵 and 𝐶𝐾 are parallel to each other and 𝐾𝐿 is a tangent to the circle at 𝐴 (i)Prove that 𝑂𝑊 is parallel to 𝐵𝐶. (ii)Prove that Angle 𝐴𝑊𝑂 = Angle 𝐴𝐾𝐶.

Page 2 Paradigm Specialising in O Level Mathematics Maths Secrets 5 The diagram shows a point 𝑃 on a circle and 𝑃𝑄 is a tangent to the circle. Points 𝐴, 𝐵 and 𝐶 lie on the circle such that 𝑃𝐴 bisects angle 𝑄𝑃𝐵 and 𝑄𝐴𝐶 is a straight line. The lines 𝑄𝐶 and 𝑃𝐵 intersect at 𝐷. (i) Prove that 𝐴𝑃 = 𝐴𝐵. (ii) Prove that 𝐶𝐷 bisects angle 𝑃𝐶𝐵. (iii) Prove that triangles 𝐶𝐷𝑃 and 𝐶𝐵𝐴 are similar. 6 The diagram shows a circle passing through points 𝐷, 𝐸, 𝐶 and 𝐹, where 𝐹𝐶 = 𝐹𝐷. The point 𝐷 lies on 𝐴𝑃 such that 𝐴𝐷 = 𝐷𝑃. 𝐷𝐶 and 𝐸𝐹 cut 𝑃𝐵 at 𝑇 such that 𝑃𝑇 = 𝑇𝐵. (i) Show that 𝐴𝐵 is a tangent to the circle at point 𝐹. (ii) By showing that triangle 𝐷𝐹𝑇 and triangle 𝐸𝐹𝐷 are similar show that 𝐷𝐹2 − 𝐹𝑇2 = 𝐹𝑇 × 𝐸𝑇. 7 Given that 𝐴𝐷 and 𝐵𝐶 are straight lines, 𝐴𝐶 bisects angle 𝐷𝐴𝑌 and 𝐴𝐵 bisects angle 𝐷𝐴𝑋, show that (i) 𝐴𝐶2 = 𝐸𝐶 × 𝐵𝐶, (ii) 𝐵𝐶 is a diameter of the circle, (iii) 𝐴𝐷 and 𝐵𝐶 are perpendicular to each other. 8 In the diagram, two circles touch each other at 𝐴. 𝑇𝐴 is tangent to both circles at 𝐴 and 𝐹𝐸 is a tangent to the smaller circle at 𝐶. Chords 𝐴𝐸 and 𝐴𝐹 intersect the smaller circle at 𝐵 and 𝐷 respectively. Prove that (i) line 𝐵𝐷 is parallel to line 𝐹𝐸, (ii) ∠𝐹𝐴𝐶 = ∠𝐶𝐴𝐸. 9 In the diagram, 𝐴𝐶𝐷𝐸 is a cyclic quadrilateral. Lines 𝐺𝐴𝐵 and 𝐹𝐸𝐻𝐶 are parallel, and line 𝐺𝐴𝐵 is a tangent to the circle at 𝐴. Lines 𝐴𝐷 and 𝐸𝐶 meet at 𝐻. Prove that (i) triangle 𝐴𝐵𝐷 and triangle 𝐶𝐵𝐴 are similar, (ii) triangle 𝐴𝐶𝐻 and triangle 𝐴𝐷𝐶 are similar, (iii) 𝐴𝐷 bisects angle 𝐶𝐷𝐸, (iv) 𝐴𝐵 × 𝐴𝐻 = 𝐴𝐶 × 𝐵𝐶.

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