year 9: loci dr j frost ([email protected]) last modified: 30 th december 2013
TRANSCRIPT
! A locus is a set of points satisfying a certain condition.
Loci
Thing A Thing B
Loci involving:
Interpretation
A given distance from point APoint
Resulting Locus
- Click to Learn
A
A given distance from line ALine - Click to
LearnA
Equidistant from 2 points or given
distance from each point.
Point Point Click to Learn
AB
Perpendicular bisector
Equidistant from 2 linesLine Line Click to
Learn
A
B
Angle bisector
Equidistant from point A and line BPoint Line Not until FP1 at
Further Maths!B
ParabolaA
?
?
?
?
?
Fixed distance from a point
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
Moo!
Fixed distance from a point
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.
3m
Click to BroshadeCommon schoolboy error: Thinking the locus will be oval in shape.
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.
Circular corners.
Straight corners.
10m
Scale: 1m : 1cm2m
2m
2m
10m
Exercise
Q1
10m
Scale: 1m : 1cm2m
10m
ExerciseI’m 2m away from the walls of a building.Copy the diagram (to scale) and draw the locus. Ensure you use a compass.
Q2
6m
6m
Click to Broshade
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?
5m
10m A
Exercise
Q3Scale: 1m : 1cm
Bonus question:What is the area of this region, is in terms of ?87.5 ?Click to Broshade
Distances from two pointsMaxi 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.
3km
Click to Brosketch
A B
5km
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.
?
Q4
AB
3km
4km
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.
Distances from two points
Q4
Click to Broshade
Equidistant from 2 pointsBut 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
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 2: Using the same distance on your compass, draw another arc, ensuring you include the points of intersection with the other 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
B
Le Problemo:Arcs don’t overlap enough, so points of intersection to draw line through is not clear.
A
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)
? ?
Equidistant from two lines
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.
A
B