Year 9 Loci Dr J Frost jfrosttiffin kingston

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

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 ? A Click to Learn Line - A given distance from line A ? Click to Learn Point Equidistant from 2 points or given distance from each point. Line Equidistant from 2 lines Point Line Equidistant from point A and line B A Perpendicular bisector ? B A A ? Angle bisector B A ? Parabola B Click to Learn Not until FP 1 at Further Maths!

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

Fixed distance from a point Common schoolboy error: Thinking the locus will be oval in shape. Click to Broshade 3 m A goat is now attached to a metal bar, by a rope of length 3 m. 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.

Exercise Q 1 I’m 2 m 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. Scale: 1 m : 1 cm 2 m 2 m Circular corners. 10 m Straight corners. 2 m 10 m

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

Exercise Q 3 Scale: 1 m : 1 cm 10 m My goat is attached to a fixed point A on a square building, of 5 m x 5 m, by a piece of rope 10 m in length. Both the goat and rope are fire resistant. What region can he reach? A 5 m Click to Broshade Bonus question: What is the area of this region, is in terms of ? 87. 5 ?

Distances from two points Q 4 Maxi is phoning his friend to get a lift to a party. He says he is 3 km away from Town A and 5 km from Town B. Sketch the locus his friend needs to check to find Maxi. 3 km A 5 km B Bonus Question: How could Maxi augment his description so the locus is just a single point? Click to Brosketch 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. ?

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

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? A STEP 2: Using the same distance on your compass, draw another arc, ensuring you include the points of intersection with the other arc. B 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. STEP 3: Your locus is the line that goes between these points of intersection. It is known as the perpendicular bisector.

Common Losses of Exam Marks A A B B Le Problemo: Arcs don’t overlap enough, so points of intersection ? to draw line through is not clear. 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) ?

Equidistant from two lines A STEP 1: Measure out some distance across each line, ensuring the distance is the same. 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