Coordinate Geometry · Learner pages

Lesson 1 · Measure distance between two points

Derive the distance formula from a right triangle.

Name or initials: ____________   Date: ____________

Horizontal and vertical coordinate changes form perpendicular legs.

1. Find the coordinate changes. From A(1, 2) to B(4, 6), the horizontal change is 3 and vertical change is 4.
2. 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.
3. Apply Pythagoras. d² = 3² + 4² = 25, so d = 5. Distance uses the nonnegative square root.
4. 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.

1. Find the distance between (1, 2) and (4, 6).

   1. 5 · 2. 7 · 3. 6

2. Find the distance between (-2, 1) and (3, 13).

   1. 17 · 2. 14 · 3. 13

3. Find the distance between (0, -1) and (6, 7).

   1. 11 · 2. 10 · 3. 14

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.

Name or initials: ____________   Date: ____________

A point partway from A to B adds the same fraction of each coordinate change.

1. 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.
2. Take one third of the changes. Both coordinate changes are 6; one third of each is 2.
3. Add to the starting point. P = (1 + 2, 2 + 2) = (3, 4).
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).

1. P divides A(0, 2) to B(15, 11) internally in AP:PB = 1:2. Find P.

   1. (5, 5) · 2. (10, 8) · 3. (15, 11)

2. P divides A(0, 2) to B(15, 11) internally in AP:PB = 2:1. Find P.

   1. (5, 5) · 2. (15, 11) · 3. (10, 8)

3. P divides A(0, 2) to B(20, 14) internally in AP:PB = 1:3. Find P.

   1. (20, 14) · 2. (5, 5) · 3. (15, 11)

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.

Name or initials: ____________   Date: ____________

A midpoint is the internal section point with equal parts.

1. Average the x-coordinates. For A(−2, 1) and B(6, 7), x of the midpoint is (−2 + 6)/2 = 2.
2. Average the y-coordinates. Its y-coordinate is (1 + 7)/2 = 4, so M = (2, 4).
3. Check equal distances. AM = √(4² + 3²) = 5 and MB = √(4² + 3²) = 5.
4. 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).

1. Find the midpoint of (1, 2) and (7, 6).

   1. (4, 4) · 2. (8, 8) · 3. (5, 5)

2. Find the midpoint of (-4, 3) and (4, 9).

   1. (0, 12) · 2. (1, 7) · 3. (0, 6)

3. Find the midpoint of (2, -3) and (12, 5).

   1. (8, 2) · 2. (7, 1) · 3. (14, 2)

Discuss: Choose two diagonals of a drawn quadrilateral. Compare their midpoints and explore when the diagonals bisect each other.

Final check · Set 1

Try one set after the lessons. Use the other set after reviewing any errors.

1. Find the distance between (-4, -3) and (5, 9).

   21 / 16 / 15

2. Find the distance between (2, -5) and (9, 19).

   26 / 25 / 31

3. P divides A(0, 2) to B(25, 17) internally in AP:PB = 2:3. Find P.

   (15, 11) / (25, 17) / (10, 8)

4. P divides A(0, 2) to B(25, 17) internally in AP:PB = 3:2. Find P.

   (25, 17) / (15, 11) / (10, 8)

5. Find the midpoint of (0, 5) and (12, 7).

   (12, 12) / (7, 7) / (6, 6)

6. Find the midpoint of (3, 1) and (5, 13).

   (5, 8) / (4, 7) / (8, 14)

Final check · Set 2

Try one set after the lessons. Use the other set after reviewing any errors.

1. Find the distance between (-1, 5) and (11, 21).

   20 / 28 / 21

2. Find the distance between (4, 0) and (24, 21).

   41 / 30 / 29

3. P divides A(0, 2) to B(25, 17) internally in AP:PB = 1:4. Find P.

   (5, 5) / (20, 14) / (25, 17)

4. P divides A(0, 2) to B(25, 17) internally in AP:PB = 4:1. Find P.

   (5, 5) / (25, 17) / (20, 14)

5. Find the midpoint of (-8, 0) and (-2, 14).

   (-5, 7) / (-10, 14) / (-4, 8)

6. Find the midpoint of (4, -4) and (12, -2).

   (16, -6) / (9, -2) / (8, -3)