Example 2 - Finding the coordinates
Four Congruent means two shapes are exactly the same shape and exactly the same size. triangles fit together to make the following shape.
The coordinates of three points A, B and C are given.
Find the coordinates of points D and E.
1. What do I have to do?
Read the question through twice. Highlight or underline the important pieces of information in the question.
2. What information do I need?
The highlighted words are the most important ones.
What is the question asking? The question wants two coordinates. The answer will be two positive coordinates.
3. What information don’t I need?
The axes are not essential to this question as it is clear from the coordinates that the positive quadrant is the one being used.
4. What maths can I do?
Step A
Use the coordinates of point A and point C to work out the height of the triangle in the bottom left-hand corner of the diagram.
A (5, 2) and C (5, 7)
The \(x\) coordinate is the same but the \(y\) coordinate has a difference of 5.
\(7 - 2 = 5\)
This means that the height of the triangle must be 5.
Congruent means the same. Therefore all four triangles when drawn the same way round must have a height of 5.
Step B
Use the coordinates of point A and point B to work out the length of the base of the triangle.
A (5, 2) and B (18, 2)
The \(y\) coordinate is the same but the \(x\) coordinate has a difference of 13.
\(18 - 5 = 13\)
The triangles are placed as shown:
Therefore the base of the triangle on the left must be 8.
\(13 - 5 = 8\)
As they are congruent, all four triangles must have a base of 8.
Each triangle therefore can be labelled.
Step C
Use the coordinates of point C to work out the coordinates of point D.
The diagram shows that to get from point C to point D the \(x\) coordinate must increase by 5 and the \(y\) coordinate must increase by 8.
\(x\) coordinate: \(5 + 5 = 10\)
\(y\) coordinate: \(7 + 8 = 15\)
Therefore the coordinates of point D are (10, 15).
Step D
Use the coordinates of point D to work out the coordinates of point E.
Point E is on the same level as point D therefore the \(y\) coordinate is unchanged.
Point E has moved 8 in the \(x\) direction therefore 8 must be added onto point D’s \(x\) coordinate.
\(x\) coordinate: \(10 + 8 = 18\)
The coordinates of point E are therefore (18, 15).
5. Is my solution correct?
It is important to check any calculations at the end.
Check that point E’s \(x\) coordinate is the same as point B’s \(x\) coordinate.
6. Have I completed everything?
The answer is supposed to be two coordinates, which it is.
Make sure the coordinates are in brackets.
Nothing else was asked for.