Shape and Space Problems Year 9  To view the problem statement, double click on the problem title. This will open up the problem in a new window. From there, you can download the full problem-and-solution Word document.

 Examples Page Ref Problem Title Objectives Ref Description Key Words 169, 183, 193 Paper Folding Similarity, Gradients A square is simply folded over and then opened out - what does the crease look like ? Similar, Slope, Gradient 185, 188, 209 Overlapping Figures Angles, Symmetry, Similar triangles How much overlap between 2 congruent regular polygons ? Congruent, Corresponding, Interior angle, Isometric 185 P Where You Like Properties of Triangles Prove that the sum of distances from any point in an equilateral triangle to the sides is constant Proof, Triangle 185, 187 L-Shape Area, Proof How to divide any 2-rectangle 'composite' in half? Symmetry, Proof, Bisect 123, 185, 247 Overlapping Squares 2 2-D Shape, Proof, Trigonometry, Surds Rotate 2 squares - how much of the lower one can you see? Simple to state, scope to explore... Congruent, Symmetry, Pythagoras, Surds, 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 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 127, 189 Truncated Square Pythagoras Another 'not-enough-information-surely?' problem that comes out nicely when Pythagoras is liberally applied ... Pythagoras, Subject of the formula 189 Double Squares Pythagoras A surprisingly simple trick to writing numbers as the sums of squares - a must-see ! Pythagoras 189, 219 Quickest Route Co-ordinate distances, Pythagoras Jeeves needs to escape across a swimming pool - but what's his quickest escape route ? Pythagoras 191, 193, 217 Triangle Dissections Congruence, Similarity, Enlargement A series of puzzles concerned with dissecting equilateral triangles into smaller parts Congruent, Similar 191 The Radius Congruence, Similarity A simple application of Similar triangles Congruent, Similar 193,61, 81, 215 Pendants Similarity, Enlargement 'Area Factors' under enlargement give quick answers to this problem Similar, Enlargement, 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, 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 200, 207 TetraCubes 3D Shape Identify shapes, then use to build mini Soma Cubes Plan, View, Symmetry 201 Sliced Cube 3-D Shape,  Pythagoras Visualize, then prove, a result about a bisected cube. Cross-Section, Plane, Properties 161, 165, 203 Graphic Convergence Co-ordinates, Mappings, Combination of Transformations A pair of mappings for x and y lead to a convergent sequence of points, which can also be viewed as a combination of transformations - quite pretty. Map, Invariant 213 Square in a Triangle Transformations, Symmetries How to squeeze the largest possible square inside any triangle? Enlargement 215, 221 Forensic Triangles Constructions, Dynamic Geometry Reconstruct the original triangle from the sides' mid-points Medians, Similar, Parallel 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 Eggs Constructions, Compasses First take an egg... Compasses, Perpendicular, Arc, Tangent 221 Triangle in a Square Constructions, Dynamic Geometry How to squeeze the largest possible equilateral triangle inside a square? Compasses, Rotation 223, 245 Height of the Tower Similar Triangles, Scale Drawing There's a tower, see, across this river, and what you've got to do is... Similarity, Elevation 133, 227 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 81, 91, 233 Average Speed Speed, Ratio + Proportionality A nice introduction to ratio methods for combining average speeds over 2 sections of a journey Speed, Average 91, 233 A Walk In The Bush Fractions, Measurements An 'average speed' problem that comes out very sweetly - involving some up and down hills Speed, Average 235 How Deep Is The Well Circles Fairly straightforward circumference calculations Pi, Circumference 237 Pie Free Circles Circles, Area A section of a circle turns out to have an area independent of Pi Pi, Arc, Pythagoras 237 Snake Eyes Circles, Pythagoras A circular area that turns out not to involve Pi ! Pi, Hypotenuse, Pythagoras 237 Target Practice Circles The middle ring of a circular target has a simple area... Pi, Proportion 237 Loo Roll Emergency Circles When the loo roll looks half-size, how much is really left? Pi, Circle, Area 35, 237 Wiggly Paths Circles Surprisingly pretty result about the area of winding pathways...(cf P.35 Garden Path) Pi, Radius, Arc 237 Square Peg, Round Hole Circles Which fits better-a square peg in a round hole, or a round peg in a...? Pi 155, 237 Pick A Shape Generate sequence, Find nth term Pick's Theorem for Areas on a dotty grid Formula, Generate Tn, Proof