A burglar is halfway up a ladder when it slips backwards at the base A burglar is halfway up a ladder when it slips backwards at the base. Show his path, as you would see it from side-on
Imagine a light fixed to the rim of a wheel. Can you imagine the locus of points that the light passes through, as you would see it when someone rode past?
Ug the caveman is riding his bike. It has square wheels. He fixes a light to the rim of the back wheel. Can you imagine the locus of points that the light passes through, as you would see it when someone rode past? And repeat!
Og the caveman is riding his bike. It has rectangular wheels. He fixes a light to the rim of the back wheel. Can you imagine the locus of points that the light passes through, as you would see it when someone rode past? And repeat!
Can you have a smooth ride with square wheels? Yes! If the road is bumpy… And if that doesn’t convince you, check out: http://www.youtube.com/watch?v=LgbWu8zJubo
Loci A locus (plural loci) is a set of points that satisfy a description. Eg find the locus of points 2cm from A 2cm A Eg find the locus of points 3cm from AB 3cm 3cm A B The standard constructions seen already can be used to find loci
Beat the teacher Find the positions in the playground where a pupil can’t be seen by the teachers marked by a black dot (buildings are represented by the grey shapes).
Regions The zoo wardens need to construct a fence 1m away from Max the Gorilla’s cage. Scale: 2cm = 1m
Any point on the bisector is equidistant from A and B Any point ‘left’ of the line is closer to A than B Any point ‘right’ of the line is closer to B than A A The perpendicular bisector is used to find: The midpoint of a line segment The set of points (a line) equidistant from two points The region closer to one point than another
The angle bisector is used to find: The set of points (a line) equidistant from two lines The region closer to one line than another B Any point on the bisector is equidistant from AB and AC A Any point ‘above’ the line is closer to AB Any point ‘below’ the line is closer to AC C
Regions Eg Find the locus of points within the triangle that are: Within 10cm of A => Draw arc, radius 10cm, centred on A Closer to B than C => Draw perpendicular bisector of BC B Anywhere ‘inside’ the arc is within 10cm of A Anywhere ‘above’ the perpendicular bisector is closer to B than C C A
Loci and regions 5cm from Z 1. Find and label the point P, inside the triangle, that is 3cm from X and 5cm from Z. Y 3cm from X P Z X 2. Shade the region, inside the quadrilateral, more than 4cm from Q Q
3. Show the set of points, inside the triangle, equidistant from A and B 4. Show the set of points, inside the triangle, equidistant from PQ and QR C P R Q
5. Shade the region, inside the quadrilateral, closer to E than F K 6. Shade the region, inside the quadrilateral, closer to LM than KL L J M
7. Shade the region outside the triangle that are no more than 2cm away
Regions satisfying 2 constraints Eg Find the locus of points within the quadrilateral that are: Within 4cm of EF Equidistant from FG and GH => Draw line 4cm away from EF => Draw bisector of angle FGH F Any point on the angle bisector is equidistant from FG and GH Anywhere ‘above’ the line is within 4cm of EF 4cm E 4cm H The locus required satisfies both descriptions G
8. Find the locus of points within the triangle that are: Within 5cm of P Closer to Q than R P R Q
9. Find the locus of points within the quadrilateral that are: More than 7cm from S Closer to VU than UT S T U V
Reaching the end of your tether yet? Mr Walker keeps his pet goat tethered to the shed in his garden, using a 4m lead. What region can the goat can reach? Scale: 2cm = 1m
Beat the teacher Find the positions in the playground where a pupil can’t be seen by the teachers marked by a black dot (buildings are represented by the grey shapes).
Loci and regions 1. Find and label the point P, inside the triangle, that is 3cm from X and 5cm from Z. Y Z X 2. Shade the region, inside the quadrilateral, more than 4cm from Q Q
3. Show the set of points, inside the triangle, equidistant from A and B 4. Show the set of points, inside the triangle, equidistant from PQ and QR C P R Q
5. Shade the region, inside the quadrilateral, closer to E than F K 6. Shade the region, inside the quadrilateral, closer to LM than KL L J M
7. Shade the region outside the triangle that are no more than 2cm away
8. Find the locus of points within the triangle that are: Within 5cm of P Closer to Q than R P R Q
9. Find the locus of points within the quadrilateral that are: More than 7cm from S Closer to VU than UT S T U V