The
Straight Line Part 2
Vocabulary
Perpendicular means to cross at right angles.
To bisect means to cut into two parts of equal shape and size.
· Perpendicular bisector
A perpendicular bisector cuts a line in half at 90 degrees.
Example
Find the equation of the perpendicular bisector
of the line joining A(0,3) and B(5,4)
· Concurrency
Lines are said to be concurrent if they pass through one common point.
Lines AB, DE and FG are all concurrent, sharing the common point O.
The perpendicular bisectors of a line are concurrent.
In a triangle, the point of intersection is called the circumcentre.
Example
Find K, the circumcentre of the triangle formed by the points
A(-5,2) , B(0,7) and C (2,0)
Steps
1) Find Midpoints
2) Find gradients
3) Solve equations to find K.
Solution
The circumcircle of a triangle
is the circle which passes through
all the vertices of the triangle and has
its centre at the circumcentre.
An
altitude of a triangle is a line drawn from a
vertex perpendicular to the opposite side. Since a triangle has three sides, it
also has three altitudes.
The
altitudes are concurrent and meet at the orthocentre
of the triangle.
Example
Triangle RST has co-ordinates
R(-5,2) , S (0,6) and T( 3,-4)
Find the equation of the altitude from S.
Solution
The
altitude from S is perpendicular to RT.
A
median of a triangle is a line from a vertex
to
the midpoint of the opposite side.
A
triangle has three medians, which cross at the centroid.
The
Centroid splits the median in the ratio 1:2
Triangle RST has co-ordinates
R(-5,2) , S (0,6) and T( 3,-4)
Find the equation of the median from R.
