10.1 Transformations II

10.1 Transformations II

10.1.1 Similarity
Two shapes are similar if
(a) the corresponding angles are equal and
(b)   the corresponding sides are proportional.

Example:

$\begin{array}{l}\angle A=\angle J={90}^{o}\\ \angle B=\angle K={50}^{o}\\ \angle C=\angle L={130}^{o}\\ \angle D=\angle M={90}^{o}\end{array}$

(All the corresponding angles are equal.)

$\begin{array}{l}\frac{AB}{JK}=\frac{5}{10}=\frac{1}{2}\\ \frac{BC}{KL}=\frac{4}{8}=\frac{1}{2}\\ \frac{CD}{LM}=\frac{2.5}{5}=\frac{1}{2}\\ \frac{AD}{JM}=\frac{3}{6}=\frac{1}{2}\end{array}$

(All the corresponding sides are proportional.)

10.1.2 Enlargement
1. Enlargement is a type of transformation where all the points of an object move from a fixed point at a constant ratio.

2. The fixed point is known as the centre of enlargement and the constant ratio is known as the scale factor.
$\text{Scale factor}=\frac{\text{length of side of image}}{\text{length of side of object}}$

3.
The object and the image are similar.

4. If A’ is the image of A under an enlargement with centre O and scale factor k, then $\frac{OA\text{'}}{OA}=k$
• if k > 0, then the image is on the same side of the object.
• if k < 0, then the image is on the opposite side of the object.
• if –1 < k < 1, then the size of the image is a reduction of the size of the object.
5. Area of image = k2 × area of object.