Geometry and measures - Properties and constructions: solve geometrical problems on coordinate axes

The idea

These questions mix shape facts with the coordinate grid. The midpoint of a segment is found by averaging the two x coordinates and averaging the two y coordinates. The length of a segment comes from its horizontal and vertical steps with a² + b² = c². The gradient is the vertical step divided by the horizontal step, and two different lines with equal gradients are parallel. Typical tasks are to show that a set of points forms a particular quadrilateral, to find a missing vertex, or to find where a line cuts an axis. Take A(1, 2) and B(7, 10). The midpoint is (4, 6). The steps are six across and eight up, so the length AB is the square root of the sum of 36 and 64, which is 10.

What earns the marks

• A correct method for a midpoint, a length or a gradient, even if the arithmetic slips.
• Coordinates written as a pair in brackets, in the right order.
• Using a shape property, such as equal gradients for parallel sides or equal lengths for a rhombus.
• A conclusion that answers the question, for example naming the quadrilateral.

Common mistakes and how to put them right

• Subtracting the coordinates instead of averaging them when finding a midpoint.

• Reading a coordinate pair the wrong way round, so that the y value is plotted along the x axis.

• Taking the horizontal step as the top of the gradient fraction.

• Counting grid squares as one unit each when the two axes are scaled differently.

Exam tips

• Sketch the points even when no grid is printed, because the shape usually shows you what to do.
• Check a midpoint by making sure it lies between the two points on your sketch.