9 Concurrence
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The document discusses concurrence of lines, which refers to three or more lines passing through a common point. It provides examples of showing lines are concurrent by: 1) Taking two line equations and solving them simultaneously to find the point of intersection. 2) Substituting the point of intersection into the remaining line(s) to prove it lies on those line(s) as well. 3) Making a statement that the lines are concurrent if they pass through the same point.
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9 Concurrence
- 1. Concurrence 3 or more lines meeting at a point 9.
- 2. Concurrence Copy the following: Intersection deals with two lines meeting at a common point. Common Point Concurrence deals with three or more lines passing through a common point. To prove that lines are concurrent: Take the equations of two of the lines to set up and solve a simultaneous equation. 2. Prove that the solution to part 1 also lies on the other line(s).
- 3. Example 1 Show that the following lines are concurrent: x + y - 9 = 0 2x + 3y = 23 3x - y = 7 Solution : Take one pair of equations and solve to find point of intersection y =5 All in form Ax + By = C - - - Sub y = 5 into x + y = 9 Point of Intersection
- 4. Example 1 Solution : 2. Show that the point you have just found also lies on the other line(s) Sub x = 4 & y = 5 in : 3. Make a statement Point of Intersection x + y - 9 = 0 2x + 3y = 23 3x - y = 7 Show that the following lines are concurrent: 3x - y LHS: RHS: AS LHS=RHS lines are concurrent
- 5. 3x – y = 7 2x + 3y = 23 x + y = 9
- 6. Homework booklet, Exercise 9
- 7. Example 2 Find k if these lines are concurrent: x + 4y = 7 3x + y = 10 x - 5y + k = 0 Solution : Take one pair of equations and solve to find point of intersection 11y =11 All in form Ax + By = C - - - Sub y = 1 into x + 4y = 7 Point of Intersection y = 1
- 8. Example 2 Solution : 2. Points ARE concurrent so substitute into 3 rd equation to find k. Sub x = 3 & y = 1 in : Point of Intersection x + 4y = 7 3x + y = 10 x - 5y + k = 0 x - 5y + k = 0 Find k if these lines are concurrent: