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GRADE 10 · A4 PRINT VIEW

Coordinate Geometry · Answer guide

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Lesson 1 · Measure distance between two points

Derive the distance formula from a right triangle.

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

    Answer: 5. Horizontal change = 3, vertical change = 4. Distance = √(3² + 4²) = √25 = 5.

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

    Answer: 13. Horizontal change = 5, vertical change = 12. Distance = √(5² + 12²) = √169 = 13.

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

    Answer: 10. Horizontal change = 6, vertical change = 8. Distance = √(6² + 8²) = √100 = 10.

Discuss: Choose two points with the same x-coordinate. Explain how the formula reduces to their vertical separation.

Teaching note: Adding coordinate changes gives a route along two streets, not usually the straight-line distance. Open responses need discussion; they are not automatically marked.

Lesson 2 · Divide a segment in a given ratio

Use the internal section formula and check which endpoint is nearer.

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

    Answer: (5, 5). P = ((1×15+2×0)/3, (1×11+2×2)/3) = (5, 5).

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

    Answer: (10, 8). P = ((2×15+1×0)/3, (2×11+1×2)/3) = (10, 8).

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

    Answer: (5, 5). P = ((1×20+3×0)/4, (1×14+3×2)/4) = (5, 5).

Discuss: Draw a segment from (0, 0) to (12, 6) and mark divisions 1:2 and 2:1. Compare their coordinates.

Teaching note: For AP:PB = m:n, the m multiplies B coordinates. Reversing the weights places the point at the opposite fractional position. Open responses need discussion; they are not automatically marked.

Lesson 3 · Use midpoint and distance checks together

Locate a midpoint and verify a segment property.

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

    Answer: (4, 4). Average each coordinate separately: ((1+7)/2, (2+6)/2) = (4, 4).

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

    Answer: (0, 6). Average each coordinate separately: ((-4+4)/2, (3+9)/2) = (0, 6).

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

    Answer: (7, 1). Average each coordinate separately: ((2+12)/2, (-3+5)/2) = (7, 1).

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

Teaching note: Many points can be equally distant from A and B. The midpoint must also lie on segment AB. Open responses need discussion; they are not automatically marked.

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).

    15. Horizontal change = 9, vertical change = 12. Distance = √(9² + 12²) = √225 = 15.

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

    25. Horizontal change = 7, vertical change = 24. Distance = √(7² + 24²) = √625 = 25.

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

    (10, 8). P = ((2×25+3×0)/5, (2×17+3×2)/5) = (10, 8).

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

    (15, 11). P = ((3×25+2×0)/5, (3×17+2×2)/5) = (15, 11).

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

    (6, 6). Average each coordinate separately: ((0+12)/2, (5+7)/2) = (6, 6).

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

    (4, 7). Average each coordinate separately: ((3+5)/2, (1+13)/2) = (4, 7).

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. Horizontal change = 12, vertical change = 16. Distance = √(12² + 16²) = √400 = 20.

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

    29. Horizontal change = 20, vertical change = 21. Distance = √(20² + 21²) = √841 = 29.

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

    (5, 5). P = ((1×25+4×0)/5, (1×17+4×2)/5) = (5, 5).

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

    (20, 14). P = ((4×25+1×0)/5, (4×17+1×2)/5) = (20, 14).

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

    (-5, 7). Average each coordinate separately: ((-8+-2)/2, (0+14)/2) = (-5, 7).

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

    (8, -3). Average each coordinate separately: ((4+12)/2, (-4+-2)/2) = (8, -3).

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.