Use gradients inside shapes on axes: parallel sides, the line joining two midpoints, the equation of a side, where two sides meet, and (Higher) perpendicular sides, heights of triangles, and the corners of squares and rhombuses.
What you will learn
- Use equal gradients to find parallel sides and the equation of a side
- Show that the line joining two midpoints of a triangle is parallel to the third side and half as long
- Find where two sides meet and the area they enclose with an axis
- Use perpendicular gradients to find a missing coordinate, a height or a diagonal (Higher)
- Find the other corners of a square from one side (Higher)
Equal gradients, parallel sides
Gradient = change in y ÷ change in x.
Two sides with the same gradient are parallel. In a parallelogram ABCD, the side CD has the gradient of AB and passes through C, which gives its equation.
- A(1, 1), B(3, 5), C(6, 4) are three corners of parallelogram ABCD
- gradient of AB = (5 − 1) ÷ (3 − 1) = 4 ÷ 2 = 2
- CD: y = 2x + c through C(6, 4): 4 = 12 + c, so c = −8
- y = 2x − 8
Join the midpoints M of AB and N of AC in any triangle ABC: MN always has the gradient of BC, and BC is twice as long. Work out M and N, then compare the steps.
The rest of this lesson and its 11 practice questions are locked.
In this lesson
- Equal gradients, parallel sides
- Where two sides meet
- Perpendicular gradients multiply to −1 (Higher)
- Squares from one side (Higher)
- Try it: First question
11 practice questions, marked as you go; about 30 minutes in all.
Key terms
Same gradient
In this unit: Graphs and coordinates: shapes on axes, regions, areas and tangents
More graphs and coordinate geometry: shapes on coordinate axes, gradients and perpendicular lines in shapes, regions with strict and inclusive boundaries, the area under a curve from trapezia, and the gradient at a point from a tangent.
- Shapes on coordinate axes
- Gradients and lines in shapes
- Regions in depth
- Area under a curve
- Gradient at a point, and the kinematics formulae
All of Graphs and coordinates: shapes on axes, regions, areas and tangents