Point A is at position 3 on a number line, point C is at position 11, and point B is between A and C at position 7. What is AB + BC, and what does this tell you?
AAB + BC = 4 + 4 = 8; this is a coincidence for this specific case
BAB + BC = 4 + 4 = 8 = AC; this confirms the Segment Addition Postulate because B lies between A and C
CAB + BC = 7 + 4 = 11; you add the position of B to the remaining distance
DAB + BC cannot be computed without knowing if A, B, C are collinear
AB = |7 − 3| = 4 and BC = |11 − 7| = 4, so AB + BC = 8 = AC = |11 − 3|. This illustrates the Segment Addition Postulate: when B lies between A and C on a line, the two partial lengths add to the total. Option D is wrong because on a number line, collinearity is given — all three points are on the same line by definition.
Question 2 Multiple Choice
A student computes the distance from point P at (1, 2) to point Q at (4, 6) as (4−1) + (6−2) = 7. What error did they make?
AThey should have computed (4+1) + (6+2) = 13 instead
BThey added the horizontal and vertical differences directly instead of using the Pythagorean theorem: d = √(3² + 4²) = 5
CThey forgot to take the absolute value of each difference before adding
DDistance in the coordinate plane cannot be computed from coordinates alone
The distance formula d = √((x₂−x₁)² + (y₂−y₁)²) comes from the Pythagorean theorem. The horizontal gap (3) and vertical gap (4) are the legs of a right triangle; the actual distance is the hypotenuse: √(9 + 16) = √25 = 5. Simply adding the differences gives the sum of the legs, not the hypotenuse — a common error that ignores the geometry underlying the formula.
Question 3 True / False
Distance between two points can be negative if the second point is to the left of the first on a number line.
TTrue
FFalse
Answer: False
Distance is always nonnegative. On a number line, distance is computed as |a − b|, and the absolute value ensures a positive result regardless of order. If A is at 8 and B is at 3, then AB = |3 − 8| = |−5| = 5, not −5. Directed distance (displacement) can be negative, but distance — which measures 'how far apart' — never can.
Question 4 True / False
The Segment Addition Postulate applies only when B lies between A and C on the same line — not merely between them in everyday language.
TTrue
FFalse
Answer: True
The geometric definition of 'between' is stricter than the everyday meaning. B is geometrically between A and C only if (1) all three points are collinear — on the same line — and (2) AB + BC = AC. If B is not on the line through A and C, you cannot use the postulate, even if B is spatially 'between' them in some loose sense. This precision matters in proofs and in problems where B's position must be established, not assumed.
Question 5 Short Answer
Why do we use the absolute value when computing distance on a number line, and why is it not needed explicitly in the coordinate-plane distance formula?
Think about your answer, then reveal below.
Model answer: On a number line, subtraction can yield a negative result depending on order (e.g., 3 − 7 = −4), but distance is always positive, so we write |a − b|. In the coordinate-plane formula, each difference is squared before taking the square root — squaring automatically makes the result nonnegative — so the absolute value is built into the squaring step.
This reveals the structural relationship between the number-line and coordinate-plane formulas: they are the same idea. The number-line version uses |a − b| to ensure positivity; the coordinate-plane version achieves the same result by squaring. Understanding this connection shows that the distance formula is not a separate rule to memorize but the Pythagorean theorem applied to coordinate differences.