Lengths of segments on number lines: Set 2

Learning Objectives Calculate the length of a segment on a number line when coordinates are given as algebraic expressions. Apply the Segment Addition Postulate to solve for unknown variables and segment lengths. Determine the coordinate of a point that partitions a segment on a number line into a given ratio. Use the definition of a midpoint to find the coordinates of endpoints or midpoints. Prove that two segments on a number line are congruent by calculating and comparing their lengths. Solve multi-step problems involving segment bisectors and algebraic expressions. Interpret absolute value as the distance between two points on a number line in formal proofs. Ever wondered how GPS calculates distances in a straight line, even with unknown starting points? 🗺️ Let's...

TermDefinitionExample Segment Addition PostulateIf three points A, B, and C are collinear and B is between A and C, then the length of segment AB plus the length of segment BC is equal to the length of the entire segment AC.If point A is at -2, B is at 3, and C is at 10, then AB = 5 and BC = 7. The postulate states AB + BC = AC, so 5 + 7 = 12. The length of AC is indeed |10 - (-2)| = 12. Midpoint of a SegmentA point that divides a segment into two congruent (equal length) segments.If M is the midpoint of segment PQ, where P is at 2 and Q is at 10, then M is at coordinate 6. The length PM = 4 and MQ = 4. Segment BisectorA point, line, ray, or another segment that intersects a segment at its midpoint, thus dividing it into two equal parts.If line *l* intersects segment AB at its midpoint M,...

Distance Formula on a Number Line d = |x₂ - x₁| To find the length of a segment, take the absolute value of the difference between the coordinates of its endpoints, x₁ and x₂. This ensures the distance is always a non-negative value. Midpoint Formula on a Number Line M = (x₁ + x₂)/2 To find the coordinate of the midpoint (M) of a segment, add the coordinates of the endpoints (x₁ and x₂) and divide by 2. This is the average of the coordinates. Segment Partition Formula (1D) P = x₁ + (m/(m+n))(x₂ - x₁) To find the coordinate of a point P that divides the directed segment from x₁ to x₂ in the ratio m:n, you add a fraction of the total segment length to the starting coordinate.