The segment joining p1(-4,-7)and p2(6,9)is divided in to four equal parts find the points of division

To find the points of division on the segment joining P1(-4,-7) and P2(6,9) that divide it into four equal parts, we need to determine the coordinates of three points.

The formula for finding a point on a line segment is given by:
P = ( (1-t) * P1 ) + ( t * P2 )

Where P is the point we want to find, P1 and P2 are the endpoints of the line segment, and t is the fraction of the distance from P1 to P2.

Since we want to divide the segment into four equal parts, we need to find the points at fractions of 1/4, 2/4, and 3/4 of the distance.

First, let's find the distance between P1 and P2 using the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
= √((6 - (-4))^2 + (9 - (-7))^2)
= √(10^2 + 16^2)
= √(100 + 256)
= √356

Now, we can find the points of division.

1. Point of division at 1/4 of the distance:
t1 = 1/4
P1/4 = ( (1 - t1) * P1 ) + ( t1 * P2 )
= ( (1 - 1/4) * (-4, -7) ) + ( 1/4 * (6, 9) )
= ( (3/4) * (-4, -7) ) + ( 1/4 * (6, 9) )
= ( (-12/4, -21/4) ) + ( (6/4, 9/4) )
= ( (-3, -5.25) ) + ( (1.5, 2.25) )
= ( -3 + 1.5, -5.25 + 2.25 )
= ( -1.5, -3 )

2. Point of division at 2/4 of the distance:
t2 = 2/4
P2/4 = ( (1 - t2) * P1 ) + ( t2 * P2 )
= ( (1 - 2/4) * (-4, -7) ) + ( 2/4 * (6, 9) )
= ( (2/4) * (-4, -7) ) + ( 2/4 * (6, 9) )
= ( (-8/4, -14/4) ) + ( (12/4, 18/4) )
= ( (-2, -3.5) ) + ( (3, 4.5) )
= ( -2 + 3, -3.5 + 4.5 )
= ( 1, 1 )

3. Point of division at 3/4 of the distance:
t3 = 3/4
P3/4 = ( (1 - t3) * P1 ) + ( t3 * P2 )
= ( (1 - 3/4) * (-4, -7) ) + ( 3/4 * (6, 9) )
= ( (1/4) * (-4, -7) ) + ( 3/4 * (6, 9) )
= ( (-4/4, -7/4) ) + ( (18/4, 27/4) )
= ( (-1, -1.75) ) + ( (4.5, 6.75) )
= ( -1 + 4.5, -1.75 + 6.75 )
= ( 3.5, 5 )

Therefore, the points of division that divide the segment joining P1(-4,-7) and P2(6,9) into four equal parts are:

P1/4 = (-1.5, -3)
P2/4 = (1, 1)
P3/4 = (3.5, 5)