Original learning companion, not full textbook exercise solutions. Covers distance, internal section and midpoint applications. Coordinate proofs of every shape property and all textbook exercises remain to be added.
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LESSON 1
Measure distance between two points
Derive the distance formula from a right triangle.
Horizontal and vertical coordinate changes form perpendicular legs.
Find the coordinate changes
From A(1, 2) to B(4, 6), the horizontal change is 3 and vertical change is 4.
Form a right triangle
A horizontal move followed by a vertical move makes legs of length 3 and 4. The direct path is the hypotenuse.
Apply Pythagoras
d² = 3² + 4² = 25, so d = 5. Distance uses the nonnegative square root.
Generalise and check order
Replacing the changes by x₂ − x₁ and y₂ − y₁ gives the formula. Reversing the points changes signs but not their squares.
Worked example: Between (−2, 1) and (1, 5), the differences are 3 and 4; distance is √(9 + 16) = 5 units. The signs of coordinates do not make the distance negative.
Watch for this: Adding coordinate changes gives a route along two streets, not usually the straight-line distance.
Discuss: Choose two points with the same x-coordinate. Explain how the formula reduces to their vertical separation.
LESSON 2
Divide a segment in a given ratio
Use the internal section formula and check which endpoint is nearer.
A point partway from A to B adds the same fraction of each coordinate change.
Interpret the ratio
P divides A(1, 2) to B(7, 8) in AP:PB = 1:2. Thus P is one third of the way from A to B.
Take one third of the changes
Both coordinate changes are 6; one third of each is 2.
Add to the starting point
P = (1 + 2, 2 + 2) = (3, 4).
Confirm the weighting
The section formula gives ((1×7 + 2×1)/3, (1×8 + 2×2)/3) = (3, 4). P is closer to A, as expected.
Worked example: Divide A(−2, 0), B(8, 10) in 2:3. P is 2/5 of the way from A, so P = (−2 + 4, 0 + 4) = (2, 4).
Watch for this: For AP:PB = m:n, the m multiplies B coordinates. Reversing the weights places the point at the opposite fractional position.
Discuss: Draw a segment from (0, 0) to (12, 6) and mark divisions 1:2 and 2:1. Compare their coordinates.
LESSON 3
Use midpoint and distance checks together
Locate a midpoint and verify a segment property.
A midpoint is the internal section point with equal parts.
Average the x-coordinates
For A(−2, 1) and B(6, 7), x of the midpoint is (−2 + 6)/2 = 2.
Average the y-coordinates
Its y-coordinate is (1 + 7)/2 = 4, so M = (2, 4).
Check equal distances
AM = √(4² + 3²) = 5 and MB = √(4² + 3²) = 5.
Check the whole segment
AB = √(8² + 6²) = 10, so AM + MB = AB. Coordinate averaging put M on the segment as well as equally distant.
Worked example: If a segment has endpoint A(2, 3) and midpoint M(5, 7), then the other endpoint is B(2×5 − 2, 2×7 − 3) = (8, 11). Averaging A and B returns (5, 7).
Watch for this: Many points can be equally distant from A and B. The midpoint must also lie on segment AB.
Discuss: Choose two diagonals of a drawn quadrilateral. Compare their midpoints and explore when the diagonals bisect each other.
