10.1 Transformations II

Posted on April 16, 2018 by user

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:

Quadrilateral ABCD is similar to quadrilateral JKLM because

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

(All the corresponding angles are equal.)

$\begin{array}{l} \frac{A B}{J K} = \frac{5}{10} = \frac{1}{2} \\ \frac{B C}{K L} = \frac{4}{8} = \frac{1}{2} \\ \frac{C D}{L M} = \frac{2.5}{5} = \frac{1}{2} \\ \frac{A D}{J M} = \frac{3}{6} = \frac{1}{2} \end{array}$

(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.

$Scale factor = \frac{length of side of image}{length of side of object}$

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

$\frac{O A'}{O A} = k$

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.