POINT OF CONCURRENCY

Example 1 :

In the figure shown below, find the concurrent lines and point of concurrency. 

Solution :

In the diagram above, since the straight lines AB, CD, PQ and RS are all meeting at the same point O, they are concurrent line.

The point of concurrency is O. 

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 : 

3x - 4y + 5 = 0 ----A

7x - 8y + 5 = 0 ----B

4x + 5y - 45 = 0 ----C

Solve A and B : 

B - 2A ----> x - 5  =  0

x  =  5

Substitute x = 5 in (1). 

3(5) - 4y + 5  =  0

15 - 4y + 5  =  0

20 - 4y  =  0

-4y  =  -20

y  =  5

The point of intersection of A and B is (5, 5). 

Now, check whether the line C satisfies the point (5, 5). 

Substitute x = 5 and y = 5 in C, 

4(5) + 5(5)  =  45 (true ?)

20 + 25  =  45 (true ?)

45  =  45 (yes, it is true)

Point of intersection of lines A and B satisfies the third line.

Therefore, all the three lines A, B and C are intersecting at the same point (5, 5) and they are concurrent. 

The point of concurrency is (5, 5). 

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