Geometry and Algebra - October 2006, Stage 4&5

Problems

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Points in Pairs

Stage: 4 Challenge Level: Challenge Level:1

In the diagram the radius length is 10 units, OP is 8 units and OQ is 6 units. If the distance PQ is 5 units what is the distance P'Q' ?

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The Line and Its Strange Pair

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

In the diagram the point P' can move to different places along the dotted line. Each position P' takes will fix a corresponding position for P. If P' moves along a straight line what does P do ?

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Mapping the Wandering Circle

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

In the diagram the point P can move to different places around the dotted circle. Each position P takes will fix a corresponding position for P'. As P moves around on that circle what will P' do?

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Intersections

Stage: 4 and 5 Challenge Level: Challenge Level:1

Change one equation in this pair of simultaneous equations very slightly and there is a big change in the solution. Why?

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Lattice Points

Stage: 5 Challenge Level: Challenge Level:1

Why are there only a few lattice points on a hyperbola and infinitely many on a parabola?

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Pick's Quadratics

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Find a quadratic formula which generalises Pick's Theorem.

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Proof of Pick's Theorem

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Follow the hints and prove Pick's Theorem.