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The Straight Line All straight lines have an equation of the form m = gradienty axis intercept C C + ve gradient - ve gradient

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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 horizontal lines have an equation of the form

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

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

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Collinearity Two lines can either be: A B C At an angle Parallel and Distinct A B C D A B C 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) 2.If the pairs of points have a point in common they MUST be collinear.

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

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Perpendicular Lines x y If we rotate the line through 90 0.

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Perpendicular Lines x y If we rotate the line through 90 0.

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Perpendicular Lines x y If we rotate the line through 90 0.

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Perpendicular Lines x y O A This is true for all perpendicular lines. If two lines with gradients m 1 and m 2 are perpendicular then Conversely, if then the lines with gradients m 1 and m 2 are perpendicular.

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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.

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

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

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1. What is the equation of the line with gradient 2 passing through the point (0,-5)? 2. Find the gradient and the y intercept of the line with equation

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Because (2,7) satisfies the equation y = 4x – 1, the point must lie on the line.

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General Equation of a Straight Line

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

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

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2. Find the equation of the line passing through P(-2,0) and Q(1,6). Using point P But what if we used point Q? Using 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.

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Lines in a Triangle 1. The Perpendicular Bisector. A perpendicular bisector will bisect a line at 90 0 at the mid point. The point of intersection is called the Circumcentre.

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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.

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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.

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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 A further point of information regarding the centroid. 2 1 The centroid is a point of TRISECTION of the medians. It divides each median in the ratio 2:1.

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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!!) F G H MEDIAN J

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

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