Measuring Segments - Interactive Activity

Definitions and Main Ideas

A segment can be measured, and its measure is the distance between its endpoints. On a number line, that distance is the absolute value of the difference of the two coordinates. On the coordinate plane, distance comes from the Distance Formula, which is the Pythagorean Theorem in disguise. Segments with equal measures are congruent, and the bar notation matters: the name of a segment and the number that measures it are different things.

Vocabulary

Measure of a Segment

The distance between the segment's endpoints. The measure of segment AB is written AB (with no bar above it): the name with the bar names the segment itself, while the name without the bar is a number. In symbols, AB names the segment itself, and AB is its measure — so a statement like AB ≈ CD says the two segments are congruent (equal in measure), while AB = CD says their measures, the numbers, are equal.

Congruent Segments

Segments that have equal measures. Two segments are congruent if and only if their measures are equal.

Distance on a Number Line

The distance between two points on a number line is the absolute value of the difference of their coordinates, so distance is never negative. If the two coordinates are a and b, the distance is written |a − b|.

Distance Formula

The distance between A(x₁, y₁) and B(x₂, y₂) in the coordinate plane is d = √((x₂ − x₁)² + (y₂ − y₁)²).

Endpoint

A point at the end of a segment or at the starting point of a ray. A segment has two endpoints.

Main Ideas

1. Congruence of segments is decided by measures: equal numbers mean congruent segments.
2. Distance is always nonnegative — take the absolute value of the difference on a number line.
3. The Distance Formula averages nothing; it combines the horizontal change and the vertical change as the legs of a right triangle.
4. Subtract coordinates in the same order in both parts of the formula, and square before adding — never add the differences and then take a square root of the sum unsquared.

Number-Line Distance Explorer

What you are looking at: A number line from −10 to 10 with two points on it, P and Q. The bold gold segment between them is segment PQ, and its measure, written PQ with no bar, is the distance from P to Q.

What to do: Drag the orange handle on point P and the purple handle on point Q along the number line. Each point stays on a whole-number mark, the two points cannot land on the same mark, and neither point can be dragged past the ends of the number line, so a handle can always be grabbed again.

What to notice: The order of the subtraction does not matter, because the absolute value makes the distance nonnegative. Drag Q to the left of P and watch: the distance stays the same for the same two marks.

Distance Formula Explorer

What you are looking at: Points A and B on a coordinate grid. The dashed lines are the legs of a right triangle whose hypotenuse is segment AB: one leg shows the horizontal change from A to B, and the other shows the vertical change. The bold gold highlight marks segment AB itself.

What to do: Drag the orange handle on point A and the purple handle on point B to new grid points. Each point snaps to a whole-number grid point, the two points cannot land on the same grid point, and dragging is limited to the grid shown, so a handle can always be grabbed again.

What to notice: The readout works the Distance Formula in six numbered steps with the actual coordinates on screen: the formula, the substitution of the coordinates, the two subtractions, the two squares, the sum under the radical, and finally the square root — exact first, then the decimal. The distance is the length of the hypotenuse, never the sum of the two legs.

Build the Distance

What to do: Each problem gives you a figure and a question about a distance. Work in three steps: (1) choose the equation to use to solve the problem, (2) carry out the computation or solve for the unknown, and (3) report the answer the question actually asks for. The diagram comes first, then the problem statement, then the step question. The bold gold highlight in the diagram always marks the part the current step asks about, and a part solved in an earlier step is labeled with its found measure. Each choice you pick animates a gold segment in the diagram to the result that choice describes, whether the choice is right or wrong, so you can see what the choice means; a missed choice is shown as a dashed amber ghost. A first miss gives a hint and another try, with the missed choice removed. A second miss on a step names the correct choice and explains it, and the activity then moves on. When all three steps are complete, the full worked solution appears. Press New Item for the next problem: the four problems take turns in order, and the problem number is shown with the step question.