The Straight Line. + ve gradient. C. C. - ve gradient. All straight lines have an equation of the form . m = gradient. y axis intercept. All horizontal lines have an equation of the form. Undefined and zero gradient.
+ ve gradient
- ve gradient
All straight lines have an equation of the form
m = gradient
y axis intercept
Consider two points on this graph.
The equation of the line is
From the diagram we can see that
Note that is the angle the line makes with the positive direction of the x axis.
Two lines can either be:
At an angle
Parallel and Distinct
Parallel and form a straight line
Points that lie on the same straight line are said to be collinear.
To prove points are collinear:
1. Show that two pairs of points have the same gradient. (parallel)
1. Prove that the points P(-6 , -5), Q(0 , -3) and R(12 , 1) are collinear.
Page 3 Exercise 1B
1. If P is the point (2,-3) and Q is the point (-1,6), find the gradient of the line perpendicular to PQ.
To find the gradient of the line perpendicular to PQ we require the
negative reciprocal of –3.
2. Triangle RST has coordinates R(1,2), S(3,7) and T(6,0). Show that the triangle is right angled at R.
Hence the triangle is right angled at R.
1. What is the equation of the line with gradient 2 passing through the point (0,-5)?
Because (2,7) satisfies the equation y = 4x – 1, the point must lie on the line.
The equation of a straight line with gradient m passing through (a,b) isFinding the equation of a Straight Line
P(x,y) is any point on the line except A.
For every position P the gradient of AP
Using point Q
Using point P
But what if we used point Q?
Regardless of the point you use the equation of the straight line will ALWAYS be the same as both points lie on the line.
1. The Perpendicular Bisector.
A perpendicular bisector will bisect a line at 900 at the mid point.
The point of intersection is called the Circumcentre.
1. A is the point (1,3) and B is the point (5,-7). Find the equation of the perpendicular bisector of AB.
To find the equation of any straight line we need a point and a gradient.
An altitude of a triangle is a line from a vertex perpendicular to the opposite side. A triangle has 3 altitudes.
The point of intersection is called the Orthocentre.
The median of a triangle is a line from a vertex to the mid point of the opposite side. A triangle has 3 medians.
The point of intersection is called the centroid
A further point of information regarding the centroid.
The centroid is a point of TRISECTION of the medians. It divides each median in the ratio 2:1.
1. F, G and H are the points (1,0), (-4,3) and (0,-1) respectively. FJ is a median of triangle FGH and HR is an altitude. Find the coordinates of the point of intersection D, of FJ and HR.
(Draw a sketch – It HELPS!!)
The point D occurs when;