Question

Show that ΔABC, where A(–2, 0), B(2, 0), C(0, 2) and ΔPQR where P(–4, 0), Q(4, 0), R(0, 2) are similar triangles

Answer 1

In ΔABC, the coordinates of the vertices are A(–2, 0), B(2, 0), C(0, 2). 

`AB = sqrt((2+2)^2 + (0 - 0)^2) = 4`

`BC = sqrt((0 - 2)^2 + (2 - 0)^2) =sqrt8 = 2sqrt2`

`CA = sqrt((0 + 2)^2 + (2 - 0)^2) = sqrt8 = 2sqrt2)`

In ΔPQR, the coordinates of the vertices are P(–4, 0), Q(4, 0), R(0, 4).

`PQ = sqrt((4+4)^2 + (0-0)^2) = 8`

`QR = sqrt((0 - 4)^2 + (4 - 0)^2) =     4sqrt2`

`PR= sqrt((0 + 4)^2 + (4 - 0)^2) = 4sqrt2`

Now, for ΔABC and ΔPQR to be similar, the corresponding sides should be proportional

So, `(AB)/(PQ) = (BC)/(QR) = (CA)/(PR)`

`=> 4/8 = (2sqrt2)/(4sqrt2) = (2sqrt2)/(4sqrt2) = 1/2`

Thus, ΔABC is similar to ΔPQR