Question

The endpoints of the directed line segment ab are a(7,4) and b(2,9). Find the coordinates of point p along segment ab so that the ration of ap:pb is 3:2.

Answer 1

To find the coordinates of point P along segment AB such that the ratio of AP:PB is 3:2, you can use the section formula. Plug in the values of A(7, 4), B(2, 9), and the ratios 3:2 into the formula to find that point P is located at (5, 6).

Step-by-step explanation:

To find the coordinates of point P along segment AB such that the ratio of AP:PB is 3:2, we can use the concept of section formula. The section formula states that if a line segment AB is divided by a point P(x, y) such that AP:PB is given, then the coordinates of point P can be found using the following formula:

P(x, y) = ( (xA × m + xB × n) / (m + n), (yA × m + yB × n) / (m + n) ),

where m and n are the given ratios (in this case, 3 and 2), and (xA, yA) and (xB, yB) are the coordinates of points A and B, respectively.

Using the given coordinates of points A(7, 4) and B(2, 9), and the ratio of 3:2, we substitute these values into the formula to find the coordinates of point P.

P(x, y) = ( (7 × 3 + 2 × 2) / (3 + 2), (4 × 3 + 9 × 2) / (3 + 2) )

P(x, y) = ( (21 + 4) / 5, (12 + 18) / 5 )

P(x, y) = ( 25 / 5, 30 / 5 )

P(x, y) = ( 5, 6 )

Answer 2

The coordinates of point P are (4, 7). This point lies along the line segment AB and divides it in the ratio of 3:2.

To find the coordinates of point P along segment AB, we can use the concept of section formula.

The section formula states that if a line segment AB is divided by a point P into two segments AP and PB in the ratio m:n, then the coordinates of point P can be found using the following formula:

Px = (m * Bx + n * Ax) / (m + n)
Py = (m * By + n * Ay) / (m + n)

In this case, we are given that the ratio of AP to PB is 3:2. So, m = 3 and n = 2.

Substituting the values into the formula, we get:

Px = (3 * 2 + 2 * 7) / (3 + 2) = (6 + 14) / 5 = 20 / 5 = 4
Py = (3 * 9 + 2 * 4) / (3 + 2) = (27 + 8) / 5 = 35 / 5 = 7