Question

Find the coordinates of the three points that divide the line segment joining P(-4, 7) and Q (10, -9) into four parts of equal length.​

Answer 1

To divide the line segment PQ into four parts of equal length, we need to find three points R, S, and T such that PR = RS = ST = TQ.

Let the coordinates of the point R be (x1, y1). Then we can express the coordinates of the other points in terms of (x1, y1) as follows:

PR = RS = ST = TQ

=> PQ/4 = PR = RS = ST = TQ/4

=> PQ = 4PR = 4RS = 4ST = 4TQ

Since PQ has endpoints P(-4, 7) and Q(10, -9), we can find its length as follows:

PQ = sqrt((10 - (-4))^2 + (-9 - 7)^2)

PQ = sqrt(196 + 256)

PQ = sqrt(452)

PQ = 2sqrt(113)

Therefore, we have:

4PR = PQ

=> PR = PQ/4

=> PR = (2sqrt(113))/2

=> PR = sqrt(113)

Using the midpoint formula, we can find the coordinates of the midpoint of PQ as follows:

Midpoint of PQ = ((x1 + 10)/2, (y1 + (-9))/2)

=> ((x1 + 10)/2, (y1 - 9)/2)

Now, we can find the coordinates of R as follows:

R = (x1, y1) = (10/3, -11/3)

Similarly, we can find the coordinates of S and T as follows:

S = ((2x1 + 10)/3, (2y1 - 9)/3) = (2, -5)

T = ((3x1 + 10)/4, (3y1 - 9)/4) = (26/7, -29/7)

Therefore, the coordinates of the three points that divide the line segment PQ into four parts of equal length are:

R = (10/3, -11/3)

S = (2, -5)

T = (26/7, -29/7)

Answer 2

A = (-1.06, 1.18)

B = (2.89, -4.15)

C = (6.84, -7.48)

Explanation:

To divide the line segment PQ into four equal parts, we need to find the coordinates of three points that divide PQ into four equal parts. Let A, B, and C be these three points such that PA = AB = BC = CQ.

We can find the coordinates of A, B, and C by finding the distances PA, AB, and BC and then using the midpoint formula to find the coordinates of the points.

First, we can find the distance between P and Q using the distance formula:

d(PQ) = sqrt((10 - (-4))^2 + (-9 - 7)^2) = sqrt(14^2 + (-16)^2) = sqrt(452) ≈ 21.26

To divide the line segment PQ into four equal parts, we need to find the length of each of the three smaller segments, which is:

d(PA) = d(AB) = d(BC) = d(CQ) = d(PQ)/4 = sqrt(452)/4 ≈ 5.32

Now, we can find the coordinates of A by starting at P and moving in the direction of Q-P for a distance of d(PA):

A = P + (d(PA)/d(PQ))(Q-P) = (-4, 7) + (5.32/21.26)(10 - (-4), -9 - 7) = (-1.06, 1.18)

Similarly, we can find the coordinates of B and C:

B = A + (d(AB)/d(PQ))(Q-P) = (-1.06, 1.18) + (5.32/21.26)(10 - (-4), -9 - 7) = (2.89, -4.15)

C = B + (d(BC)/d(PQ))(Q-P) = (2.89, -4.15) + (5.32/21.26)(10 - (-4), -9 - 7) = (6.84, -7.48)

Therefore, the coordinates of the three points that divide the line segment joining P(-4, 7) and Q (10, -9) into four parts of equal length are:

A = (-1.06, 1.18)

B = (2.89, -4.15)

C = (6.84, -7.48)