Question

If three distinct points \(P, Q,\) and \(R\) all lie on a line, and if \(d(P, Q)=d(Q, R),\) then \(Q\) is called the _____ of the line segment from \(P\) to \(R\).

Short answer

Midpoint

Step-by-step solution

Understand the Definition

Read the problem statement carefully. We need to identify what special name is given to point Q if it lies on the line segment from P to R and the distances from P to Q and Q to R are equal.

Identify the Midpoint Concept

Recall the definition of a midpoint. The midpoint of a line segment is the point that divides the segment into two equal lengths. So, if point Q divides the line segment PR into two segments of equal length, Q is the midpoint of the line segment from P to R.

Apply the Concept to the Problem

According to the problem, if d(P, Q) = d(Q, R), Q must be the midpoint of the line segment PR because it divides it into two equal lengths.

Key concepts

Line Segment

A line segment is a portion of a line that has two distinct end points. These end points mark the limits of the segment, defining its length. Unlike a line that extends infinitely in both directions, a line segment has a finite length. One can think of a line segment as a piece or a snippet of an entire line.

If you have points P and R on a line, the line segment is the part of the line that starts at P and ends at R. All the points between P and R, including P and R, make up the line segment PR. It is important to understand that every point on a line segment is exactly located between the two end points.

One practical way to visualize a line segment is to think of it like a stick or a strip with a fixed start and end. This concept is fundamental in geometry and helps in understanding various properties of shapes and forms.

Equal Length

Equal length means that two or more line segments have the same measure. If you were to measure these segments with a ruler, they would all show the same value. In our exercise, the lengths PQ and QR are described as being equal, which is a crucial clue.

When we say that two segments are of equal length, it implies they are congruent. Consequently, when point Q is equally distant from P and R, it is not just any point but a special one called the midpoint. This is because equal lengths from a single point segment the line into identical halves.

Understanding the concept of equal length helps in dividing, analyzing, and comparing different parts of a geometric figure. This ensures that calculations and inferences are accurate and coherent.

Distance

In geometry, distance refers to the space between two points. The distance can be calculated using various methods depending on the coordinates of the points. For a basic line segment, the distance between two points P and Q can simply be measured with a ruler or calculated if the coordinates are known.

For our problem, the distance d(P, Q) = d(Q, R), which means that the point Q is equally spaced between P and R. This equal spacing is what defines Q as the midpoint. Calculating distances accurately is critical in geometry as it helps determine relationships between points and shapes.

Distance is often the first step in more complex geometric problems, such as finding the center of a circle, locating a specific point on a shape, or determining the perimeter of a polygon. It provides a foundational understanding that supports many other concepts in mathematics.