Question

ΔABC undergoes a dilation by a scale factor. Using the coordinates of ΔABC and ΔA'B'C', prove that the triangles are similar by AA.

Answer 1

Given:

ΔABC undergoes a dilation by a scale factor and comes as ΔA'B'C'.

To show that both the triangles are similar.

Formula

• By the condition of similarity we get,

If two triangles have three pairs of sides in the same ratio, then the triangles are similar.

• By Pythagoras theorem we get,

`Hypotenuse^2 = Base^2+Height^2`

Now,

In ΔABC,

AB = 18 unit

BC = 10 unit

So,
`AC^2 = AB^2+BC^2`

or,
`AC^2 = 18^2+10^2`

or,
`AC = √(424)`

Again,

In ΔA'B'C'

A'B' = 9 unit

B'C' = 5 unit

So,
`A'C' ^2 = A'B'^2+B'C'^2`

or,
`A'C'^2 = 9^2+5^2`

or,
`A'C' = √(106)`

Now,

`(AB)/(A'B') = (18)/(9) = 2`

`(BC)/(B'C') = (10)/(5) = 2`

`(AC)/(A'C') = (√(424) )/(√(106) ) = 2`

Hence,

All the ratios are equal.

Therefore, we can conclude that,

ΔABC and ΔA'B'C' are similar.