Problem 17 - The Calculus with Analytic Geometry

Question

Find the coordinates of the three points that divide the line segment from \(A(-5,3)\) to \(B(6,8)\) into four equal parts.

Step-by-Step Solution

Answer

The coordinates of the points are (-2.25, 4.25), (0.5, 5.5), and (3.25, 6.75).

Step 1 - Identify the endpoints

The coordinates of point A are \(A(-5, 3)\) and the coordinates of point B are \(B(6, 8)\).

Step 2 - Determine the total intervals

Since the line segment is divided into four equal parts, there will be three points that divide the segment into four parts.

Step 3 - Calculate the increments

Calculate the increments in both the x and y coordinates. \(x_{increment} = \frac{(6 - (-5))}{4} = \frac{11}{4} = 2.75\) and \(y_{increment} = \frac{(8 - 3)}{4} = \frac{5}{4} = 1.25\).

Step 4 - Find the 1st dividing point

To find the first dividing point, add the increments to point A. \(P_1=(-5 + 2.75, 3 + 1.25) = (-2.25, 4.25)\).

Step 5 - Find the 2nd dividing point

To find the second dividing point, add the increments twice to point A. \(P_2=(-5 + 2 \times 2.75, 3 + 2 \times 1.25) = (0.5, 5.5)\).

Step 6 - Find the 3rd dividing point

To find the third dividing point, add the increments three times to point A. \(P_3=(-5 + 3 \times 2.75, 3 + 3 \times 1.25) = (3.25, 6.75)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, helps us understand geometric figures using a coordinate system. It's an essential tool in mathematics because it allows us to represent shapes and solve geometric problems with algebra. In this exercise, we use the coordinate system to find specific points on a line segment. The points are defined by their coordinates \((x, y)\). By knowing the coordinates of two endpoints of a line segment, we can find the coordinates of points that divide this segment into equal parts.

Increments Calculation

Increments calculation involves determining the distance between points along the x and y axes. This is done by finding the change in the x and y coordinates between the two endpoints of a line segment. Here’s how:

- For the given points, A \((-5, 3)\) and B \((6, 8)\), we first find the difference in their coordinates: \(x_{increment} = \frac{(6 - (-5))}{4} = 2.75\).
- Similarly, for the y coordinates: \(y_{increment} = \frac{(8 - 3)}{4} = 1.25\).

By calculating these increments, we can understand how much we need to move along the x and y axes to reach our desired points.

Equal Parts Division

Dividing a segment into equal parts means finding points along the segment that split it into equally spaced divisions. To do this:

- We add increments to the starting point A multiple times.
- For example, the first dividing point, \((P_1)\), is found by adding the increments once: \(P_1 = (-5 + 2.75, 3 + 1.25) = (-2.25, 4.25)\).
- The second point, \((P_2)\), is found by adding twice the increments: \(P_2 = (-5 + 2 \times 2.75, 3 + 2 \times 1.25) = (0.5, 5.5)\).
- The third point, \((P_3)\), is calculated by adding three times the increments: \(P_3 = (-5 + 3 \times 2.75, 3 + 3 \times 1.25) = (3.25, 6.75)\).

This method ensures that we divide the line segment into four equal parts accurately.