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# 9 Concurrence

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### 9 Concurrence

1. 1. Concurrence 3 or more lines meeting at a point 9.
2. 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: <ul><li>Take the equations of two of the lines to set up and solve a </li></ul><ul><li>simultaneous equation. </li></ul>2. Prove that the solution to part 1 also lies on the other line(s).
3. 3. Example 1 Show that the following lines are concurrent: x + y - 9 = 0 2x + 3y = 23 3x - y = 7 Solution : <ul><li>Take one pair of equations and </li></ul><ul><li>solve to find point of intersection </li></ul>y =5 All in form Ax + By = C - - - Sub y = 5 into x + y = 9 Point of Intersection
4. 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. 5. 3x – y = 7 2x + 3y = 23 x + y = 9
6. 6. Homework booklet, Exercise 9
7. 7. Example 2 Find k if these lines are concurrent: x + 4y = 7 3x + y = 10 x - 5y + k = 0 Solution : <ul><li>Take one pair of equations and </li></ul><ul><li>solve to find point of intersection </li></ul>11y =11 All in form Ax + By = C - - - Sub y = 1 into x + 4y = 7 Point of Intersection y = 1
8. 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: