CONCURRENT LINES

If three or more non parallel lines meet at the same point, then the lines are said to be concurrent lines and the point at where the lines meet is called point of concurrency. 

In the figure given below, the straight lines AB, CD and EF meet at the point P.

Here, P is the point of concurrency. 

Example 1 :

From the figure given below, find out the concurrent lines and point of concurrency.

Solution :

The straight lines AB, CD, PQ and RS are concurrent lines. These lines are passing through the point O. 

Therefore O is the point of concurrency. 

Example 2 :

Prove that the following lines are concurrent and find the point of concurrency.

3x - 4y + 5 = 0

7x - 8y + 5 = 0

4x + 5y = 45

Solution : 

Plan to solve the problem :

Solve any two of the given three equations to find the point of intersection of the selected lines. Substitute the point of intersection into the third equation and check, whether it satisfies the third equation. If the point of intersection satisfies the third equation, then the lines are concurrent and the point of intersection of the first two lines is the point of concurrency.

3x - 4y + 5 = 0 ---- (1)

7x - 8y + 5 = 0 ---- (2)

4x + 5y = 45 ---- (3)

Solve (1) and (2) to get the point of intersection.

2(1) - (2) :

-x = -5

x = 5

Substitute x = 5 into (1).

3(5) - 4y + 5 = 0

15 - 4y + 5 = 0

-4y + 20 = 0

-4y = -20

y = 5

The point of intersection of (1) and (2) is (5, 5).

Substitute (5, 4) into (3).

4(5) + 5(5) = 45  

20 + 25 = 45

45 = 45 (true)

Point of intersection (1) and (2) satisfies (3).

Therefore, the given three lines are concurrent and the point of concurrency is (5, 5).

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