Question

Triangle ABC is being enlarged using a scale factor of ½ and centre (1,9) to give triangle A'B'C'. a) What are the coordinates of the vertex C'? b) What is the length of the side A'B'?

Answer 1

By rigid transformation, we get the following results:

a) The coordinates of vertex C' is (3, 5). b) The length of the side A'B' is 3 units.

How to determine the image of a side and a vertex by rigid transformation

Herein we find the representation of triangle ABC, whose vertices are shown in the figure. The coordinates of the vertices are: A(x, y) = (5, 5), B(x, y) = (11, 5), C(x, y) = (5, 1). We need to determine the coordinates of the images of the vertices by a rigid transformation known as dilation. Dilation is defined by following expression:

P'(x, y) = O(x, y) + k · [P(x, y) - O(x, y)]

Where:

• O(x, y) - Center of dilation.
• k - Scale factor
• P(x, y) - Original point
• P'(x, y) - Resulting point.

If we know that O(x, y) = (1, 9), A(x, y) = (5, 5), B(x, y) = (11, 5) and C(x, y) = (5, 1), then the images of the vertices are, respectively:

A'(x, y) = (1, 9) + 0.5 · [(5, 5) - (1, 9)]

A'(x, y) = (1, 9) + 0.5 · (4, - 4)

A'(x, y) = (1, 9) + (2, - 2)

A'(x, y) = (3, 7)

B'(x, y) = (1, 9) + 0.5 · [(11, 5) - (1, 9)]

B'(x, y) = (1, 9) + 0.5 · (10, - 4)

B'(x, y) = (1, 9) + (5, - 2)

B'(x, y) = (6, 7)

C'(x, y) = (1, 9) + 0.5 · [(5, 1) - (1, 9)]

C'(x, y) = (1, 9) + 0.5 · (4, - 8)

C'(x, y) = (1, 9) + (2, - 4)

C'(x, y) = (3, 5)

And the length of side A'B' is:

`r = √((6 - 3)^2 + (7 - 7)^2)`

r = 3