Question

Three segment measures are given. The three points named are collinear. Determine which point is between the other two. \(X Z=36, Y Z=17, X Y=19\)

Short answer

Point Y is between X and Z.

Step-by-step solution

Understanding the Problem

We need to determine which point is between the other points given the segment measures. The points X, Y, and Z are on a straight line (collinear), and the segment measures are provided as follows: \(XZ = 36\), \(YZ = 17\), and \(XY = 19\).

Analyzing Possible Configurations

Since the line is straight, there are only two possible configurations for the points. Configuration 1: \(X\) is leftmost, followed by \(Y\), then \(Z\) (X--Y--Z). Configuration 2: \(X\) is leftmost, followed by \(Z\), then \(Y\) (X--Z--Y).

Calculating Total Lengths

In configuration X--Y--Z, the total length would be \(XY + YZ = 19 + 17 = 36\). In configuration X--Z--Y, the total length would be \(XZ + ZY\) or \(36 + 17 = 53\), which does not match the given total \(XZ = 36\).

Checking Consistency with Given Data

For the configuration X--Y--Z, the total length equals the given \(XZ = 36\). Therefore, this configuration is consistent with the data (X--Y--Z). In this arrangement, \(Y\) is between \(X\) and \(Z\).

Final Conclusion

Thus, after checking both possibilities, \(Y\) is confirmed to be between \(X\) and \(Z\).

Key concepts

Segment Measures

Understanding segment measures is essential in geometry problem solving. Segments are parts of a line that connect two points. In our problem, segment measures are the lengths of the line segments formed by the points \(X\), \(Y\), and \(Z\). These measures are given as follows:

• \(XZ = 36\)
• \(YZ = 17\)
• \(XY = 19\)

These values tell us how far apart each pair of points is on a straight line. Knowing the measures allows us to hypothesize about the positions of the points relative to each other, which is a vital step in solving the problem. The consistency of these measures helps verify the configuration of the points on the line.

Point Configuration

Determining the configuration of points is key in understanding their positions on the line. Collinear points mean they all lie on the same straight line. For the given problem, we have three points; \(X\), \(Y\), and \(Z\), and we need to find out which point lies between the other two.

Possible configurations are crucial to consider:

• Configuration 1: \(X\)---\(Y\)---\(Z\). Here, \(Y\) is between \(X\) and \(Z\).
• Configuration 2: \(X\)---\(Z\)---\(Y\). In this scenario, \(Z\) would be between \(X\) and \(Y\).

Each configuration leads to different interpretations of the segment measures. By adding segments \(XY\) and \(YZ\), and checking if it equals \(XZ\), we test each hypothesis. For Configuration 1, the sum \(19 + 17 = 36\) matches \(XZ\), revealing the correct point configuration: \(X\)---\(Y\)---\(Z\).

Geometry Problem Solving

Geometry problem solving often involves logical reasoning and hypothesis testing, as demonstrated in this problem. Initially, we start by thoroughly understanding the problem statement. The given segment measures guide us through forming hypotheses about possible point configurations.

Next, using the measurements, we validate each configuration by calculating the total lengths. This straightforward subtraction and addition allow us to compare the results with the given data. Testing these possibilities systematically helps eliminate incorrect options.

• Key Strategy: Calculate and verify against known segment lengths.
• Check Consistency: Ensure that calculated total length matches the given segment measure.

Applying this method ensures accuracy, leading us to a logical conclusion and reinforcing our problem-solving skills. Every step builds a clear path toward finding which point is truly between the others.