Question

Three segment measures are given. The three points named are collinear. Determine which point is between the other two. \(R S=45, T S=19, R T=26\)

Short answer

Point \(T\) is between points \(R\) and \(S\).

Step-by-step solution

Understand the Concept

When three points are collinear, they all lie on the same straight line. One point will be between the other two. The sum of the distances from the between point to the other two points will be equal to the total distance between the outer points.

List Segment Relations

We are given three measurements: \(R S = 45\), \(T S = 19\), and \(R T = 26\). Each of these is a segment on the same line. We need to determine which configuration allows these measurements.

Analyze the Sum of Distances

Check the sums of each pair to see if they match the sum of the other segment. If \(S\) is between \(R\) and \(T\), then \(R S + T S = R T\). Substitute the known values: \(45 + 19\). This does not equal \(26\).

Test Different Configurations

Test if \(T\) is between \(R\) and \(S\), then \(R T + T S = R S\). Substitute the known values: \(26 + 19 = 45\). This is correct, so \(T\) is between \(R\) and \(S\).

Conclude the Correct Configuration

Based on the satisfied condition from Step 4, we determine that the point \(T\) is between points \(R\) and \(S\).

Key concepts

Segment Measures

Understanding segment measures is essential for solving geometry problems involving collinear points. When we talk about segment measures, we are referring to the lengths or distances between two points on a line segment. In this exercise, we are given three segment measures:

• \(R S = 45\)
• \(T S = 19\)
• \(R T = 26\)

Given these measures, our task is to determine how these points are positioned relative to each other. Knowing these distances allows us to paint a clearer picture of the configuration of the three points on the line. By analyzing these segment measures, we can infer which point might be lying between the other two, playing a crucial role in solving the problem.

Geometry Problem Solving

Geometry problem solving involves breaking down a geometry problem into understandable parts. In the case of collinear points, we follow logical steps to find relationships and verify conditions. This process starts with understanding fundamental concepts, such as what it means for points to be collinear and recognizing the significance of a midpoint.
When three points are collinear, they share the same line. Our task is to determine which point lies between the other two. Through the problem, one effective technique is analyzing the sum of segment distances.

• If one point is between two other points, the sum of the distances from that point to the other points equals the distance between the outermost points.
• In this problem, we tried different assumptions to see which one fits – checking if \(S\) or \(T\) is between the other two points.

Logical testing and verification should be methodical and organization skills in geometry problem solving are just as important as intuitive thinking. By applying these, we can reach the correct conclusion confidently.

Distance Calculation

Distance calculation forms the backbone of many geometry problems involving lines and collinear points. The primary goal of this exercise was to use the given distances to determine the arrangement of points. Distance calculation is straightforward when we substitute the provided segment values into different possible configurations to verify which satisfactorily meets the condition of collinearity.
In the given exercise, we calculated:

• If \(S\) was between \(R\) and \(T\), we would expect \(RS + TS = RT\). However, \(45 + 19 eq 26\), so \(S\) cannot be between \(R\) and \(T\).
• Alternatively, we tested if \(T\) is between \(R\) and \(S\). Here, it means \(RT + TS = RS\). Substituting the values, we find that \(26 + 19 = 45\), confirming \(T\) is indeed between \(R\) and \(S\).

Through accurate distance calculations, we gain essential insights into the spatial relationships among the points, proving that understanding how to compute and verify distances is integral to mastering geometry problems.