Seema placed a lighedbulb at point O on the ceiling and directly below it placed a table. Now, she put a cardboard of shape ABCD between table and lighted bulb. Then a shadow of ABCD is casted on the table as A'B'C'D' (see figure). Quadrilateral AB'C'D' in an enlargement of ABCD with scale factor 1:2, Also, AB = 1.5 cm, BC = 25 cm, CD = 24 cm and AD = 2.1 cm; ZA = 105?, ZB = 100?, ZC = 70? and ZD = 85. (a) What is the measurment of angle A'? (i) 105? (ii) 100? (b) What is the lenght of A'B' ? (iii) 70? (iv) 80? (i) 1.5 cm (ii) 3 cm (iii) 5 cm (iv) 2.5 cm (c) What is the sum of angles of quadrilateral A'B'C'D'? (i) 180? (ii) 360? (iii) 270? (iv) None of these (d) What is the ratio of sides A'B' and A'D' ? (i) 5:7 (ii) 7:5 (iii) 1:1 (iv) 1:2 (e) What is the sum of angles of C' and D' ? (i) 105? (ii) 100? (iii) 155? (iv) 140?
Activity 1 : Place a lighted bulb at a
point 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
ABCD (see Fig.6.4).
Note that 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'.
Here, you can also note that 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' + AB'B.
C' C and D' D. By actually measuring the angles and the sides of the two
quadrilaterals, you may verify that
(1) ZA= ZA, ZB = LB', Z C = C, D = Z D' and
AB BC CD DA
A'B' B'C' C'D' D'A'
This again emphasises that two polygons of the same number of sides are
similar. if (i) all the corresponding angles are equal and (ii) all the corresponding
sides are in the same ratio (or proportion).