Question

Three segment measures are given. The three points named are collinear. Determine which point is between the other two. \(A B=32, B C=13.8, A C=18.2\)

Short answer

Point A is between B and C.

Step-by-step solution

Understand the Problem

We have three collinear points, A, B, and C. The segment measures given are AB = 32, BC = 13.8, and AC = 18.2. Our goal is to determine which point is between the other two on this line.

Recall the Collinearity Property

When three points are collinear, the distance from the first point to the third point is the sum of the distances of the segments between the first-to-second and second-to-third points. Therefore, if B is between A and C, then AB + BC should equal AC.

Try All Possible Scenarios

Evaluate each possible order of points to find which one satisfies the segment property of collinearity. 1. Assume B is between A and C: Calculate AB + BC and compare with AC. 2. Assume A is between B and C: Calculate BC + AC and compare with AB. 3. Assume C is between A and B: Calculate AC + BC and compare with AB.

Check for B between A and C

Calculate:\[ AB + BC = 32 + 13.8 = 45.8 \]Since 45.8 ≠ 18.2, B cannot be between A and C.

Check for A between B and C

Calculate:\[ BC + AC = 13.8 + 18.2 = 32 \]Since 32 = 32, A can be between B and C.

Verify Other Possibility

Calculate:\[ AC + BC = 18.2 + 13.8 = 32 \]Compare it with AB value:Since 32 = 32, C should be the endpoint in segment AC or BC as well. However, for this configuration, B is closer to one end.

Key concepts

Segment Measure

Understanding segment measures is crucial when dealing with geometry problems like collinearity. Segment measure refers to the length of a straight line between two points. When you have multiple measurements, the key is to identify relationships or conditions between these line segments.

Let's consider three segments: \(AB=32\), \(BC=13.8\), and \(AC=18.2\). These measurements tell us how long each segment is. When trying to figure out which point is between the others, your task is to determine which pair of segments can combine to match the measure of another segment.

• For example, if \(AB + BC = AC\), then point B is between A and C.
• If \(BC + AC = AB\), point A is between B and C.
• And if \(AC + AB = BC\), then C is between A and B.

This way, segment measures guide you in solving the collinearity problem.

Collinear Points

Collinear points are points that lie on the same straight line. This is a fundamental concept in geometry because it helps determine how different segments connect.

To determine if points A, B, and C are collinear, you use the property that the distances between them can be summed linearly. For example, given the segment measures \(AB=32\), \(BC=13.8\), and \(AC=18.2\), you're tasked with confirming if these points lie on the same line, which they appear to do since the exercise states they are collinear.

To solve this, check which arrangement of points satisfies the linear sum equation:

• If \(AB + BC \equiv AC\), B is between A and C.
• If \(BC + AC \equiv AB\), A is between B and C.
• If \(AC + AB \equiv BC\), C is between A and B.

Understanding collinear points involves checking these linear relationships.

Geometry Problem Solving

Solving geometry problems involves strategic analysis and logical deduction. Initially, it's about understanding the concepts and then applying them to find a solution. With our problem, we have to use the segment measures to determine collinearity to decide point arrangement.

The first step in problem-solving is to verify each possible scenario using your understood concepts. Calculating the sum of segments for each condition allows you to find the right answer. Here's how you might approach this:

• Consider each point as potentially being between the others.
• Solve for each configuration, applying the collinearity property to see which fits.
• Compare the calculated sums with known segment measures to verify your hypothesis.

By systematically evaluating each potential scenario, you eliminate incorrect answers and find the right arrangement. This practice helps build a stronger understanding of geometry.