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MathidiaMathematics Academy
Coordinates, graphs and transformations
Lesson 67 of 75
Easy practice

Coordinate problems

Objective. Use properties of shapes to determine missing vertices.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Coordinate problems use the properties of shapes, such as equal sides or parallel lines, to find a missing vertex when the other vertices are already known. For a rectangle or parallelogram, opposite sides are equal and parallel, so the horizontal and vertical shifts between known vertices can be applied to find the missing one. For example, if three corners of a rectangle are (1,1), (5,1) and (5,4), the fourth corner must be (1,4), keeping the same horizontal shift as the opposite side. Sketching the shape on a grid often makes the missing coordinate obvious.

Worked example

Three vertices of a rectangle are A(2,3), B(7,3) and C(7,8). Find the coordinates of the fourth vertex D.

Diagram of a rectangle
  1. 1Identify the horizontal shift between A and B: from x=2 to x=7, a shift of 5.
  2. 2Identify the vertical shift between B and C: from y=3 to y=8, a shift of 5.
  3. 3Since D must align with A horizontally and with C vertically, use x=2 and y=8.
  4. 4Check the sides: AD and BC should both be vertical sides of length 5.

Answer: D = (2, 8)

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Common mistake. Pupils sometimes guess a missing vertex without checking it keeps the shape's sides parallel and equal in length. Always check the horizontal and vertical shifts match the opposite side of the shape.

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