Question

Show that ΔABC with vertices A(–2, 0), B(0, 2) and C(2, 0) is similar to ΔDEF with vertices D(–4, 0), E(0, 4) and F(4, 0).

Answer 1

Given: A(–2, 0), B(0, 2), C(2, 0) and D(–4, 0), E(0, 4), F(4, 0).

Step-wise calculation:

1. Notice DEF is obtained by multiplying each coordinate of ABC by 2:

2 × A = 2(–2, 0)

= (–4, 0)

= D

2 × B = 2(0, 2)

= (0, 4)

= E

2 × C = 2(2, 0)

= (4, 0)

= F 

So, DEF is a dilation of ABC about the origin with scale factor 2.

Hence, corresponding angles are equal.

2. Compute side lengths of ΔABC:

`AB = sqrt(0 - (-2)^2 + (2 - 0)^2` 

= `sqrt(4 + 4)`

= `sqrt(8)`

= `2sqrt(2)`

`BC = sqrt((2 - 0)^2 + (0 - 2)^2)` 

= `sqrt(4 + 4)`

= `sqrt(8)`

= `2sqrt(2)`

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

 = `sqrt(16)`

= 4

3. Compute side lengths of ΔDEF:

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

= `sqrt(16 + 16)`

= `sqrt(32)`

= `4sqrt(2)`

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

= `sqrt(16 + 16)`

= `4sqrt(2)`

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

= `sqrt(64)`

= 8

4. Compare ratios (or observe factor 2):

`(AB)/(DE) = (2sqrt(2))/(4sqrt(2))`

= `1/2`

`(BC)/(EF) = (2sqrt(2))/(4sqrt(2))`

= `1/2`

`(CA)/(FD) = 4/8`

= `1/2`

All corresponding side ratios are equal (common ratio `1/2`), so sides are proportional. This verifies similarity by the SSS similarity criterion.

ΔABC ~ ΔDEF with correspondence A ↔ D, B ↔ E, C ↔ F. The similarity is a dilation about the origin with scale factor 2 (equivalently the side-length scale factor from ABC to DEF is 2).