1. LOCI
Introduction to

2. Introduction to Loci What is a Locus?
Definition: A Locus is the Path Traced by an object or point, that moves under certain conditions or rules.
Locus of Boat = Path traced by the boat as it follows required route.
Boat =
Object
Required Route =
Rules / Conditions

3. Real Life Examples
Path Traced?
Rules?

4. Prior Knowledge of Topic
Construction of Lines
Geometric Constructions; Bisect the line, Bisect the Angle
Plane Figures; The Circle, Triangles
Where have we encountered Loci already?
Circle

5. Can have multiple locus from the
Circle:
A circle is the locus / path of all points a set distance (radius) from a fixed point called the centre. 5m
Tree Example
Plant new tree, 5m from centre point of existing tree
Can have multiple locus from the
same point.

6. Locus of all points a set distance from a line or object.
illustrate method using
set square and chalk of certain length.
Locus of all points a set
Distance from edge of
football pitch

7. Bisector of a line:
The bisector of a line is the locus/path of all points which are an equal distance from the end points of a line. X A B Y
Halfway Line X,Y is the locus of all points, equidistant from the penalty
Spots.
Centreline of road.

8. Bisector of an Angle:
The bisector of an angle is the locus/path of all points which are an equal distance from the sides of an angle.
The distances are measured perpendicular to each arm
of the angle.

9. Triangle concurrency theorems
2. The Mediators (i.e. perpendicular bisectors) of the sides of a triangle are concurrent.
The point of intersection of the mediators of the sides of a triangle is called the circumcentre.
As |OC| = |OB| = |OA|, the circumscribed circle can be drrawn about the vertices of the triangle.

10. Triangle concurrency theorems
1. The bisectors of the angles of a triangle are concurrent.
The point of intersection of the bisectors of the angles of a triangle is called the Incentre.

11. Triangle concurrency theorems
3. The perpendiculars from the vertices of a triangle to the opposite sides are concurrent.
The point of intersection of the perpendiculars from the vertices of a triangle to the opposite sides is called the orthocentre.

12. Triangle concurrency theorems
4. The medians of a triangle are concurrent
The median of a triangle is a line segment joining a vertex to the midpoint of the side opposite that vertex.
The point of intersection of the medians of a triangle is called the centroid.

13. Basketball moving through air
The Parabola
The parabola is the locus of all points that are the same distance from a fixed line called the directrix and a fixed point called the focus.
Basketball moving through air
forms parabolic curve.

14. The Ellipse
The ellipse is the locus of a point, the sum of whose distance from two fixed points (focal points) is constant.
i.e. the sum of the distances from the pencil tip to the pins (focal points) is constant.