Download Lesson 4: Triangle Centres

Lesson 4: Triangle Centres
Triangle Properties:
•
The sum of the three interior angles in a triangle is always 180 degrees.
•
An equilateral triangle has three equal sides and thus angles; the measure of each angle in
an equilateral triangle is 60 degrees.
•
•
The longest side of a triangle is always opposite the largest angle.
The shortest side of a triangle is always opposite the smallest angle.
•
In an isosceles triangle, the angles opposite the equal sides are equal.
Concurrent Lines – two or more lines that intersect at a point.
1. ALTITUDE – the perpendicular distance from a vertex to the opposite side of a triangle
C
Altitude
B
E
D
Orthocentre – the point of intersection of the three altitudes of a triangle
PROPERTIES:
The orthocenter lies:
- inside an acute triangle
- outside an obtuse triangle
- on a vertex of a right angled
triangle.
B
Orthocentre
C
D
2. MEDIANS
Midpoint – a point that divides a line segment into two equal parts.
Midpoint, M divides line segment AB into two equal parts
A
M
B
Median – a line that joins a vertex to the midpoint of the opposite side of a triangle
A
C
Median
B
Centroid – the point of intersection of the three medians of a triangle
PROPERTIES:
The centroid
- Is the centre of mass of the triangle
because it is the point at which the
triangle can be balanced.
- divides each median in the ratio 1:2
Centroid
3. PERPENDICUALR BISECTORS
Perpendicular Bisector – a line that is perpendicular to a line segment and divides it into two equal parts.
A
Perpendicular Bisector
B
C
M
Circumcentre – the point of intersection of the tree perpendicular bisectors of a triangle
Circumcentre
PROPERTIES:
The circumcentre lies:
Inside an acute triangle
Outside an obtuse triangle
On a side of a right angled triangle
It is the centre of a circle that passes
through the vertices of a triangle
Circumcircle or circumscribed circle
Angle Bisector – a line separating an angle into two equal parts
B
Angle Bisector
D
C
Incentre – the point at which the three bisectors of the vertex angles of a triangle intersect.
γ
γ
Incircle
α
α
Incentre
PROPERTIES:
The incentre is a centre of a circle
inside the triangle that meets each side
at exactly one point.
** For equilateral triangles, ALL of the centres (orthocenter, centroid, cicumcentre and incentre) are in the
SAME location **
HOMEWORK:
Text: p. 398 # 2, 3 and Worksheet: Triangle Centres #1-4