Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Given \(A(5,-8)\) and \(B(-6,2)\), find the point on segment \(A B\) that is three- fourths of the way from \(A\) to \(B\).

Step-by-Step Solution

Answer

The point is \((-3.25, -0.5)\).

Step 1: Find the Differences in Coordinates

First, compute the differences between the coordinates of points \(A\) and \(B\). The differences are calculated as follows: for the \(x\)-coordinates, \(-6 - 5 = -11\), and for the \(y\)-coordinates, \(2 - (-8) = 10\).

Step 2: Determine the Weighted Differences

We need the point that is three-fourths of the way from \(A\) to \(B\). Multiply the differences by \(\frac{3}{4}\): \(x\text{-difference} = -11 \times \frac{3}{4} = -8.25\), and \(y\text{-difference} = 10 \times \frac{3}{4} = 7.5\).

Step 3: Calculate the New Coordinates

Starting from point \(A(5, -8)\), add the weighted differences to find the new point. The new \(x\)-coordinate is \(5 + (-8.25) = -3.25\). The new \(y\)-coordinate is \(-8 + 7.5 = -0.5\).

Step 4: Present the Solution

The coordinates of the point three-fourths of the way from \(A\) to \(B\) are \((-3.25, -0.5)\).

Key Concepts

Segment Division

Assume you have a line segment connecting two points, and you want to find another point along this segment. This is precisely what segment division is about.
It involves dividing the segment into specific proportional parts. In our case, the task is to find a point three-fourths of the way from one endpoint to another. This process is applicable in various fields, such as engineering, design, and geography.

• Define two points, say point A and point B, on a plane.
• Determine the ratio in which you want to divide the segment AB.
• The ratio dictates how far the new point will be from the starting point (e.g., three-fourths).

By applying these simple steps, dividing any segment proportionally becomes straightforward. The concept primarily focuses on the positional relationship between the split parts of the segment.

Linear Interpolation

Linear interpolation is a method to determine a value within two known values on a straight line. If you have two endpoints, say A and B, you need to find a point within this segment at a specific ratio.
This concept is crucial in calculus, computer graphics, and data analysis.

• Begin with the coordinates of the two known points.
• Calculate the difference between the x-values and y-values of the endpoints.
• Multiply these differences by the desired ratio.

The resulting values from this multiplication are added to the x and y coordinates of the starting point to find the interpolated value.

Midpoint Formula

While the midpoint formula is not directly used in segment division by a specific ratio like three-fourths, it's essential to grasp when learning about coordinate geometry. The midpoint formula helps to locate the midpoint of a line segment between two points.
This is generally useful for determining equidistant points in various segments. The formula is written as:\[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]The midpoint is merely the average of the x and y coordinates of the endpoints.

• It's used to find the center between two distinct points, A and B.
• The result is a balanced point equidistant from both A and B.

While not needed for specific ratios other than one-half, understanding the midpoint formula broadens your grasp of coordinate operations.

Coordinate Calculation

Coordinate calculation involves finding precise points on a Cartesian plane. Using the formula that integrates both the x and y-axis differences with a desired ratio can simplify finding new points.
For our exercise, this resulted in computing the new point by:

• Calculating the differences between corresponding coordinates of the two known points.
• Applying a weight or ratio to these differences to find the proportionate point along the segment.
• Adjusting the initial coordinates by these weighted differences to establish the new position.

Ultimately, this coordinates-based approach is used to determine precise positions in engineering tasks and GIS systems. It enhances accuracy by mathematically determining positions along straight lines on the plane.