Exercises: Enlargements with a centre

Find the image of the point $A(2, 3)$ under an enlargement with centre the origin and scale factor $3$.

Find the image of the point $B(4, -2)$ under an enlargement with centre the origin and scale factor $\frac{1}{2}$.

Find $|3 + 4i|$, the modulus of the complex number $3 + 4i$.

Find the image of the point $P(5, 4)$ under an enlargement with centre $C(1, 2)$ and scale factor $2$.

Triangle $ABC$ has vertices $A(1, 1)$, $B(3, 1)$ and $C(3, 4)$. It is enlarged with centre the origin and scale factor $2$.

1. a) Find the coordinates of $A'$, $B'$ and $C'$.
2. b) Find the ratio of the area of $A'B'C'$ to the area of $ABC$.

Given $z_1 = 5 - 12i$ and $z_2 = -6 + 8i$:

1. a) Find $|z_1|$ and $|z_2|$.
2. b) Which of the two points representing $z_1$ and $z_2$ on the Argand diagram is closer to the origin?
3. c) Find $|z_1 + z_2|$.

An enlargement maps $A(2, 1)$ to $A'(8, 7)$ and has scale factor $3$.

1. a) Find the coordinates of the centre of enlargement.
2. b) Find the image of $B(0, 4)$.
3. c) Triangle $ABC$ has area $5\text{ cm}^2$. State the area of its image and explain why.

A rectangle with vertices $(1, 1)$, $(4, 1)$, $(4, 3)$ and $(1, 3)$ is enlarged with centre $(0, 0)$ and scale factor $k$. The image has perimeter $30$ units.

1. a) Find the perimeter of the original rectangle.
2. b) Find $k$.
3. c) Find the area of the image and the coordinates of its vertices.

The complex number $z = a + 4i$ has modulus $5$, where $a$ is a real number.

1. a) Find the two possible values of $a$.
2. b) Describe the set of all complex numbers $w$ with $|w| = 5$ on the Argand diagram.
3. c) Show that if $|z| = 5$ then $|2z| = 10$, and relate this to an enlargement of the Argand diagram with centre the origin.