Year 9: Loci

1. Year 9: Loci Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 30th December 2013

2. Loci ! A locus is a set of points satisfying a certain condition. Loci involving: Thing A Thing B Interpretation Resulting Locus ? Point - A given distance from point A Click to Learn A ? Line - A given distance from line A Click to Learn A ? Point Point Equidistant from 2 points or given distance from each point. Click to Learn Perpendicular bisector A B ? Line Line Equidistant from 2 lines Click to Learn A Angle bisector B ? Point Line Equidistant from point A and line B Not until FP1 at Further Maths! A Parabola B

3. Fixed distance from a point Moo! A goat is attached to a post, by a rope of length 3m. Shade the locus representing the points the goat can reach. 3m Click to Broshade

4. Fixed distance from a point Common schoolboy error: Thinking the locus will be oval in shape. Click to Broshade 3m A goat is now attached to a metal bar, by a rope of length 3m. The rope is attached to the bar by a ring, which is allowed to move freely along the bar. Shade the locus representing the points the goat can reach.

5. Exercise I’m 2m away from the walls of a building. Where could I be? Copy the diagram (to scale) and draw the locus. Ensure you use a compass. Q1 Scale: 1m : 1cm 2m 2m Circular corners. 10m Straight corners. 2m 10m

6. Exercise I’m 2m away from the walls of a building. Copy the diagram (to scale) and draw the locus. Ensure you use a compass. Q2 Scale: 1m : 1cm 2m 6m 10m 6m 10m Click to Broshade

7. Exercise Q3 Scale: 1m : 1cm My goat is attached to a fixed point A on a square building, of 5m x 5m, by a piece of rope 10m in length. Both the goat and rope are fire resistant. What region can he reach? A 10m Bonus question: What is the area of this region, is in terms of ? 87.5  5m Click to Broshade ?

8. Distances from two points Maxi is phoning his friend to get a lift to a party. He says he is 3km away from Town A and 5km from Town B. Sketch the locus his friend needs to check to find Maxi. Q4 5km 3km A B Bonus Question: How could Maxi augment his description so the locus is just a single point? He just needs a third landmark to describe his distance from. The process of determining location using distances from points is known as trilateration, and is used for example in GPS. It is often confused with triangulation, which uses angles to determine location rather than distances. ? Click to Brosketch

9. Distances from two points A goat is at most 3km from A and at least 4km from B. Shade the resulting locus representing the region the goat can be in. Q4 3km 4km A B Click to Broshade

10. Equidistant from 2 points But now suppose we don’t have a fixed distance from each point, but just require the distance from both points to be the same. What is the locus now? STEP 2: Using the same distance on your compass, draw another arc, ensuring you include the points of intersection with the other arc. A B STEP 3: Your locus is the line that goes between these points of intersection. It is known as the perpendicular bisector. STEP 1: Put your compass on A and set the distance so that it’s slightly more than halfway between A and B. Draw an arc.

11. Common Losses of Exam Marks A A B B Le Problemo: Locus is not long enough. (Since it’s actually infinitely long, we want to draw it sufficiently long to suggest it’s infinite) Le Problemo: Arcs don’t overlap enough, so points of intersection to draw line through is not clear. ? ?

12. Equidistant from two lines STEP 1: Measure out some distance across each line, ensuring the distance is the same. A STEP 2: The locus is just the perpendicular bisector of these two points. The line is known as the angle bisector because it splits the angle in half. B