# Polygons on the coordinate plane · Year 10

> Polygons on the coordinate plane, Year 10: Areas and more. Explanation, worked examples and free practice for the national curriculum in England.

In Polygons on the coordinate plane (Year 10) you practise 3 skills: Triangles from their vertices · Areas · Quadrilaterals from their vertices.

When the vertices of a polygon are given on a grid, the coordinates tell you almost everything: the lengths of the sides, the perimeter, the area, even the kind of polygon. The length of a segment comes from Pythagoras, using the difference in the x coordinates and the difference in the y coordinates.

The gradient of a line is the change in y divided by the change in x. Lines are parallel when their gradients are equal, and perpendicular when the gradients multiply to minus one. That is how you recognise a parallelogram, a rectangle or a trapezium, and check whether a triangle has a right angle.

For an area, choose a side parallel to one of the axes: its length and the height to it are differences of a single coordinate. When the sides are given as equations, each vertex is where two of the lines meet.

**Tip:** Plot the points on a grid before calculating. The sketch shows which side is parallel to an axis and where to start.

## What you learn

### Triangles from their vertices

- Example: The vertices of the triangle are A(2, −7) , B(5, −3) and C(2, 1) . (A triangle on a grid, its vertices marked with letters) What is the perimeter of the triangle?
- Answer: 18
- Solution: The squares of the sides, by Pythagoras: AB² = 3² + 4² = 25 , so AB = 5 , BC² = 3² + 4² = 25 , so BC = 5 , AC = 8 . The perimeter is 5 + 5 + 8 = 18 .

### Areas

- Example: The vertices of the triangle ABC are A(−3, 5) , B(−1, 8) and C(−6, 8) . (A triangle on a grid, its vertices marked with letters) What is the area of the triangle?
- Answer: 7.5
- Solution: The side BC is parallel to the x axis, and its length is 5 . The height to it is |5 − 8| = 3 . The area is 5 × 3 ÷ 2 = 7.5 .

### Quadrilaterals from their vertices

- Example: The vertices of the quadrilateral ABCD are A(−2, 2) , B(−4, −4) , C(1, −6) and D(3, 0) . (A quadrilateral on a grid, its vertices marked with letters) What is the most precise name for the quadrilateral?
- Options: A rectangle · A trapezium · None of these · A parallelogram
- Answer: A parallelogram
- Solution: The gradients of the sides, in order: m₁ = 3 , m₂ = −2/5 , m₃ = 3 and m₄ = −2/5 . Both pairs of opposite sides are parallel: m₁ = m₃, m₂ = m₄ . The gradients of two adjacent sides multiply to m₁ × m₂ = −6/5 , not −1 . So there is no right angle, and the quadrilateral is a parallelogram.

The exercises are made by rules and the answers are worked out in code. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).

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