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 Examples Page Ref Problem Title Objectives Ref Description Key Words 185, 188, 209 Overlapping Figures Angles, Symmetry, Similar triangles How much overlap between 2 congruent regular polygons ? Congruent, Corresponding, Interior angle 123, 185, 247 Overlapping Squares 2 2-D Shape, Proof, Pythagoras, Trigonometry Rotate 2 squares - how much of the lower one can you see? Simple to state, scope to explore... Congruent, Symmetry, Pythagoras, Surds, Proof 187, 189 Parallel Squares 2-D Shape, Proof, Symmetry, Congruence Draw squares on the sides of any parallelogram, and join their centres... what have you got? Congruent, Symmetry, Proof 187, 199 A Conic Journey Pythagoras, 2-D and 3-D Shape Find the shortest route across a rather special cone. Pythagoras, Cross-section, Chord 187, 189, 199 A Taut Rope 3-D Shape, Pythagoras The old Fly walking around the cuboid problem, redressed (Alternative Nets). Pythagoras, Net 189 The Right Plot Pythagoras A rather special plot of land - quite easy. Pythagoras 189 Construct A Root Pythagoras, Squares Use ruler and compasses to construct the square root of 29 - several ways... Pythagoras, Constructions, Surds 91, 127, 189 The Magic of Pythagoras Pythagoras This Problem is a variation on the Well in the Courtyard problem... and has a lovely, surprising answer as unwanted terms 'cancel out' ! Pythagoras, Subject of the formula 189, 219 Quickest Route Co-ordinate distances, Pythagoras Jeeves needs to escape across a swimming pool - but what's his quickest escape route ? Pythagoras 189 Hero Triangles Pythagoras Putting two Pythagorean triples together to form an integer-area Hero triangle Pythagoras 193 Lego Triangles Similarity Making similar triangles from 6 pieces of Lego Ratio, Similar 193 Quad Parks Similarity, Proof Show that the midpoints of any quadrilateral form a parallelogram Proof, Similar 193 Diagonal Cuts Similarity, Proof Similar triangles show how a diagonal line is trisected Proof, Similar 193 Inner Triangle Similarity, Proof Using similar triangles to find the largest area of a triangle drawn within a triangle Proof, Similar 193 Diamond Ring Similar Triangles Find the size of a circle inside a Rhombus Similar 193, 235, 237 Shortest Half 2-D Shape, Dynamic Geometry What is the most efficient way to divide an equilateral triangle into two equal areas ? Arcs, Sectors, Scale Factor 193, 215, 234 Sculpt Big Ratio, Enlargement, Similarity, Cylinders A sculptor chooses between similar large and small designs, in order to maximize profits... Scale factor, Volume, Enlarge, Proportional 193 Intersecting Chords Similarity, Angles in a Circle The simplest of theorems using 'equal angles on an arc' Similar, Corresponding, Arc, Chord 193 Tunnel Vision Similar Triangles Find the width of a tunnel given the size of the truck going through it Similar 193 Paper Crease Similar Triangles Find the length of the crease when you fold a piece of paper over Similar 193 Circle Transversals Similarity, Angles in a Circle An interesting property of a circle and two lines, with an unexpected degree of freedom in the construction... Arc, Similar, Construction 195, 235, 247 Polygon Pi Approximations Polygons, Circles, Trigonometry Using Archimedes' method of Regular Polygons to find increasingly good estimates of Pi - plus some Trial and Improvement practice. Limit, Sin, Tan, Opposite etc Arc, Chord, Pi 197 The Security Cameras Angles In A Circle Angles at the centre of a circular Art gallery room - an application of the circle Theorems! Arc, Circles 197 Four Corners Angle in a Semi-Circle Two rectangles overlap - can you see which groups of points lie on common circles? Semicircle, Circles 197 CircumCircle Angles In A Circle Finding the radius of a circumcircle turns out to be a treat Arc, Circles 197, 221 Altitudes and Orthocentres Constructions, Loci, Dynamic Geometry A 'dynamic geometry' investigation Constructions, Locus, Perpendicular 201 Facing Up To Football 1 Properties of 3-D shape Use the unfamiliar idea of angle sums at a vertex to deduce how many vertices an Icosahedron  has, and extend...! Vertex, Angle 201 Facing Up To Football 2 Properties of 3-D shape Use the now familiar idea of angle sums at a vertex to deduce how many faces a Truncated Icosahedron (football)  has, and extend...! Vertex, Angle 155, 201, 281 Ringing The Changes 2-D Representation of 3-D Shape Working through permutations of 4 'bells', using systematic sequencing. Has a very beautiful solution, modelled as the vertices of a truncated Octahedron ! Plane projection, Vertex, Edge 213 Paper Sizes Ratio, Enlargement A0, A1 etc, then into 3D for a 'Golden Cuboid' Scale factor, Enlarge, Proportional 219 Points In Between Co-ordinates Finding mid-points and points of trisection, using 'weighted average' co-ordinates, and hence finding the 'centroid' Co-ordinates, Line, Mid-point, Average, Graphs 221, 227 Centroids Triangles, Constructions, Loci, Dynamic Geometry A 'dynamic geometry' investigation Constructions, Locus, Median 221, 227 Angle Puzzle Find Locus, Constructions As the angles of a triangle vary, find the minimum area Constructions, Locus, Proof 221 Circle Tangents Circles, Dynamic Geometry, Proof An unexpected property of circles and their tangents Constructions, Tangents, Inscribed circle 223, 245 Height of the Tower Similar Triangles, Scale Drawing, Trigonometry There's a tower, see, across this river, and what you've got to do is... Trigonometry, Similarity, Elevation 225 The Out-of-Town Store Loci, Constructions Find the point within any triangle which minimises the total distance from each vertex...( the Fermat point) Constructions,  Proof, Locus 227, 197 The Down and Out Sponge Loci, Constructions What is the area that a semicircular sponge wipes in the corner of a window ? Locus,  Angles In A Circle, Region 227, 133, 235 Overlapping Squares 1 Loci, Graphs, 2-D Shape This problem offers an element of surprise in that the locus of possible solutions isn’t the straight line that pupils may well expect Region, Proof, Locus 233, 91 A Walk In The Bush Fractions, Measurements An 'average speed' problem that comes out very sweetly - involving some up and down hills Speed, Average 239, 120 Simpson's Rule and the Volume of a Sphere Volumes This exercise leads the pupil through applications of Simpson’s rule to the formula for the volume of a sphere. It is intriguing that all these volumes are given exactly by the rule. Right prisms, Volume, 237 Target Practice Circles The middle ring of a circular target has a simple area... Pi, Proportion 237 What Size are the Cylinders Volumes and Enlargement Two similar cylinders come from a block of material - how big is each? Ratio, Enlargement, Volume, Scale 239 Candy Floss Volume of Cylinders How long is a thread of Candy Floss spun from a block of sugar? Volume 247 Square in a Triangle Trigonometry What's the largest square that can be drawn inside a regular triangle ? This problem would benefit from the use of the Sine Rule. Sin, Hypotenuse etc