Question

The line segment joining the points (3, -4) and (1, 2) is trisected at the points P and Q. If the coordinates of P and Q are (p, -2) and (5/3, q) respectively. Find the values of p and q.

Answer 1

We have two points A (3,−4) and B (1, 2). There are two points P (p,−2) and Q(5/3, q) which trisect the line segment joining A and B.

Now according to the section formula if any point P divides a line segment joining `A(x_1,y_1)` and `B(x_2, y_2)` in the ratio m: n internally than,

`P(x,y) = ((nx_1 + mx_2)/(m + n)"," (ny_1 + my_2)/(m + n))`

The point P is the point of trisection of the line segment AB. So, P divides AB in the ratio 1: 2

Now we will use section formula to find the co-ordinates of unknown point A as,

`P(p, -2) = ((2(3) +  1(1))/(1 + 2), (2(-4) + 1(2))/(1 + 2)))`

`= (7/3, -2)`

Equate the individual terms on both the sides. We get,

`p = 7/3`

Similarly, the point Q is the point of trisection of the line segment AB. So, Q divides AB in the ratio 2: 1

Now we will use section formula to find the co-ordinates of unknown point A as,

`Q(5/3, q) = ((2(1) + 1(3))/(1 + 2)), ((2(2) + 1(-4))/(1 + 2))`

`= (5/3,0)`

Equate the individual terms on both the sides. We get,

q = 0