The Straight Line

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 Gradient is a measure of slope. If a line has zero gradient it has zero slope. A line with zero slope is horizontal. Consider two points on this graph. The equation of the line is

All vertical lines have an equation of the form Consider two points on this graph. (undefined) The equation of the line is

y From the diagram we can see that x Note that  is the angle the line makes with the positive direction of the x axis.

C C B B B D A C A A Collinearity 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) • If the pairs of points have a point in common they MUST be • collinear.

1. Prove that the points P(-6 , -5), Q(0 , -3) and R(12 , 1) are collinear. Page 3 Exercise 1B

y x Perpendicular Lines If we rotate the line through 900.

y Perpendicular Lines x If we rotate the line through 900.

y x Conversely, if then the lines with gradients m1 and m2 are perpendicular. Perpendicular Lines A O This is true for all perpendicular lines. If two lines with gradients m1 and m2 are perpendicular then

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.

R 2. Triangle RST has coordinates R(1,2), S(3,7) and T(6,0). Show that the triangle is right angled at R. T S Hence the triangle is right angled at R.

All straight lines have an equation of the form y m A(0,C) x Equation of a Straight Line P(x,y) is any point on the line except A. For every position P the gradient of AP is

2. Find the gradient and the y intercept of the line with equation 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.

General Equation of a Straight Line

The equation of a straight line with gradient m passing through (a,b) is Finding the equation of a Straight Line • To find the equation of a straight we need • A Gradient • A Point on the line

y m A(a,b) x P(x,y) is any point on the line except A. For every position P the gradient of AP

2. Find the equation of the line passing through P(-2,0) and Q(1,6). 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.

Lines in a Triangle 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.

2. The Altitude. 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.

3. The Median of a Triangle. 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 1 2 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.

G H F 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!!) MEDIAN J

R G J H F ALTITUDE H(0,-1) D The point D occurs when;

The Straight Line