Q.1 If a, b,c and u ,v ,w are the complex numbers representing the vertices of two triangles such that c= (1- r) a +rb and w =(1-r)u +rv, where r is a complex number, then the two triangles
(a) same area
(b) are similar
(c) are congruent
(d) none of these
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2 Answers
62One approach to think could be.....
r(b-a) makes an angle with b-a which is same as what r(v-u) makes with v-u
Can you guess why? It has to do with rotation property of complex numbres...
also the ratio of the sides is of magnitude (b-a) is to (v-u) (one pair of sides..)
62Another approach could be to think of shifting the whole thing to origin in both these cases seperately...
so we will replace a, b, by 0, b-a
and u, v by 0, v-u
so the new values will look like
c=r(b-a)
w=r(v-u)
so the three vertices of the triangle will be
0, (b-a), r(b-a)
and
0, (v-u), r(v-u)
Now i think this is very simple to think because we can further think of b-a as x and v-u as y
so the vertices of first triangle are
0, x, rx
and of 2nd triangle are
0, y, ry
Now what conclusion can you make?
