Department of Mathematics University of Leicester Equation of a Line Department of Mathematics University of Leicester

Parallel and Perpendicular Lines Contents Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines

Parallel and Perpendicular Lines Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Introduction We are used to the equation for a straight line. We’ll look at where this equation comes form, how to form it and what we can do with it once we’ve got it. We are also going to see alternative ways of writing the same equation. Next

Parallel and Perpendicular Lines Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Forming the Equation Suppose we have: How do we get the y-value from the x-value? It’s a straight line, so the ratio between x and y values is the same for all values. Eg. For this line, , or . Ratio = = gradient, or m. We get . Next

Parallel and Perpendicular Lines Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines See what happens if you vary the value of m. 1 2 3 -1 -2 -3 Next

Parallel and Perpendicular Lines Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Forming the Equation What if the line doesn’t go through the origin? If the line starts at : Then it’s , but with c added on to all the y-values. So . c Next

Parallel and Perpendicular Lines Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines See what happens if you vary the value of c. 2 4 6 -2 -4 -6 Next

Parallel and Perpendicular Lines Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Forming the Equation But if we’re not starting at anymore, we can’t just find by finding the ratio of the values. So instead, we use the ratio of the difference in value: Next

Parallel and Perpendicular Lines Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Forming the Equation If the line slopes down instead of up, the change in x will be negative: So the gradient will be negative. Next

Parallel and Perpendicular Lines Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Try plotting an equation 2 4 6 -2 -4 -6 Next

Parallel and Perpendicular Lines Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Forming the Equation What is the equation of this line: 1 2 3 -1 -2

Parallel and Perpendicular Lines Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Forming the Equation What is the equation of this line: 1 2 3

Forming the Equation: Alternative ways of writing it Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Forming the Equation: Alternative ways of writing it Question: What is the equation of the line joining and ? Answer: Start with , Then we know , so , so , or Next

Forming the Equation: Alternative ways of writing it Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Forming the Equation: Alternative ways of writing it You could rewrite with everything on the same side of the equation: And then if we have fractions, multiply through by the denominators to get Eg. Next

Forming the Equation: Alternative ways of writing it Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Forming the Equation: Alternative ways of writing it Which of the following lines joins and ? All of them

Using the equation: Intersection Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Using the equation: Intersection If two lines are not parallel then they will have exactly one point of intersection. You can find this by letting the 2 lines have the same - and -values, so they become 2 simultaneous equations… eg. The point of intersection is (-3,-3). Next

Using the equation: Intersection Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Using the equation: Intersection Where do and intersect? Solve the simultaneous equations:

Using the equation: Distance Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Using the equation: Distance You can find the distance between two points using Pythagoras’s theorem: eg. Find the distance between (1,-2) and (3,1). We get: Then distance = hypotenuse= . (1,-2) (3,1) Next

Using the equation: Distance Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Using the equation: Distance What is the distance between the points (3,6) and (-4,10)?

Using the equation: Midpoint Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Using the equation: Midpoint To find the midpoint of two points, you just find the average of the -coordinate and the average of the -coordinate. So the midpoint of and is: Next

Using the equation: Midpoint Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Using the equation: Midpoint What is the midpoint of (5,7) and (-3,11)?

Parallel and Perpendicular Lines Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Parallel and Perpendicular Lines Parallel lines will have the same gradient, because for the same change in , they both have the same change in . So . Next

Parallel and Perpendicular Lines Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Parallel and Perpendicular Lines For perpendicular lines, , and . So . Next

Parallel and Perpendicular Lines Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Parallel and Perpendicular Lines What is the gradient of the line parallel to ? Next

Parallel and Perpendicular Lines Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Parallel and Perpendicular Lines Which of the following lines is perpendicular to ?

Parallel and Perpendicular Lines Introduction Forming the Equation Using the Equation Parallel and Perpendicular Lines Conclusion A straight line has the equation: , or m = gradient, c = y-intercept We can find the Intersection Point of 2 lines, and also the Distance or Midpoint between 2 points. For parallel lines, . For perpendicular lines, . Next