6.2 Similar Figures

Activity 1 :  Place a lighted bulb at a point O on the ceiling  and directly below it a table in your classroom. Let us cut a polygon, say a quadrilateral ABCD, from a plane cardboard and place this cardboard parallel to the ground between the lighted bulb  and the table. Then a shadow of ABCD is cast on the table. Mark the outline of this shadow as A′B′C′D′

The quadrilateral A′B ′C′D′  is an enlargement (or magnification) of  the quadrilateral ABCD. This is because of the property of light that light propogates in a straight line. You may also note that A′ lies on ray OA, B′ lies on ray OB, C′ lies on  OC and D′ lies on OD.  Thus, quadrilaterals A′B′C ′D ′  and ABCD are of the same shape but of different sizes.

So, quadrilateral  A′B ′C′D′  is similiar to quadrilateral ABCD. We can also say that quadrilateral ABCD is similar to the quadrilateral A′B ′C’D’.

Vertex A′ corresponds to vertex A, vertex B′ corresponds to vertex B, vertex C′ corresponds to vertex C and vertex  D′ corresponds to vertex D. Symbolically, these correspondences are represented as A′ ↔ A, B′ ↔ B, C′ ↔ C and D′ ↔ D. By actually measuring the angles and the sides of the two quadrilaterals, you may verify that

i)$$\angle A=\angle A’,  \angle B=\angle B’, \angle C=\angle C’, \angle D=\angle D’$$ and

ii)$$\frac{AB}{A’B’}=\frac{BC}{B’C’}=\frac{CD}{C’D’}=\frac{DA}{D’A’}$$

(i) all the corresponding angles are equal and (ii) all the corresponding sides are in the same ratio (or proportion).

quadrilaterals ABCD and PQRS of are similar.

Remark : verify that if one polygon is similar to another polygon and this second polygon is similar to a third polygon, then the first polygon is similar to the third polygon.

the two quadrilaterals (a square and a rectangle) , corresponding angles are equal, but their corresponding sides are not in the same ratio.

the two quadrilaterals are not similar. Similarly, corresponding sides are in the same ratio, but their corresponding angles are not equal. Again, the two polygons (quadrilaterals) are not similar.