Question

2. Line segment AB has endpoints at A(-9 , 3) and B(1, 8).

We want to find the coordinates of point P so that P partitions AB into a part-to-whole ratio of 1 : 5.

a. What will the fraction m/n equal in our formula?

Answer 1

The coordinates of point P that partitions line segment AB into a ratio of 1:5 are found using a specific formula. In this formula, the fraction m/n is represented by the ratio 1/5, which reflects the division of the segment into 1 part and 5 parts respectively.

To find the coordinates of point P that partitions line segment AB into a part-to-whole ratio of 1:5 at points A(-9 , 3) and B(1, 8), we use the formula given by:

x = (x1 + k*x2) / (1 + k)
y = (y1 + k*y2) / (1 + k).

In this case, k represents the division ratio for the partition, which is 1:5. The point P will thus divide the line segment AB in a way where AP is one unit and PB is five units long.

For the ratio of 1:5, k=1/5, so the formula becomes:

x = (x1 + (1/5)*x2) / (1 + (1/5))
y = (y1 + (1/5)*y2) / (1 + (1/5)).

This results in the formula transforming to:

x = (-9 + (1/5)*1) / (1 + (1/5))
y = (3 + (1/5)*8) / (1 + (1/5)).

To find the fraction m/n in our formula, we look at the ratios placed within the calculations of x and y. Here, the 1/5 ratio used in both x and y calculations represents the fraction m/n.

Answer 2

Therefore, the coordinates of point P are (-22/3, 23/6).

Explanation:

To find the coordinates of point P that partitions line segment AB into a part-to-whole ratio of 1:5, we can use the concept of section formula.

The section formula states that if a line segment AB is divided by a point P such that AP:PB = m:n, then the coordinates of point P can be found using the following formula:

Px = (n * Ax + m * Bx) / (m + n)

Py = (n * Ay + m * By) / (m + n)

In this case, the part-to-whole ratio is 1:5, so m = 1 and n = 5.

Substituting the given coordinates of points A and B into the formula, we get:

Px = (5 * (-9) + 1 * 1) / (1 + 5)

= (-45 + 1) / 6

= -44 / 6

= -22/3