Show that `Delta ABC`, where `A(-2, 0), B(2, 0), C(0,2) and DeltaPQR` where `P(-4, 0), Q(4,0) and R(0,4)` are similar triangles.

Explanation

We have, `A-=(-2, 0), B-=(2, 0) and C-=(0, 2)`
` therefore` Length of sides of `DeltaABC` are,
`" "AB=sqrt((2+2)^(2)+(0-2)^(2))=4" "`…(1)
`" "B C=sqrt((0-2)^(2)+(2-0)^(2))=2sqrt(2)" "...(2)`
`" "CA=sqrt((0+2)^(2)+(2-0)^(2))=2sqrt(2)" "...(3)`

Also, it is given that
`" "P-=(-4, 0), Q-=(4, 0) and R-=(0,4)`
`therefore` Length of sides of `DeltaPQR` are
`" "PQ=sqrt((4+4)^(2)+(0-0)^(2))=8" "...(4)`
`" "QR=sqrt((0-4)^(2) +(4-0)^(2))=4sqrt(2)" "...(5)`
`" "PR=sqrt((0+4)^(2)+(4-0)^(2))=4sqrt(2)" "...(6)`
Here, `" "(AB)/( PQ)=(4)/(8)=(1)/(2)" "`[from (1) and (4)]
`" "(BC )/(QR)=(2sqrt(2))/(4sqrt(2 ))=(1)/(2)" "`[from (2) and (5)]
and `" "(AC)/(PR)=(2sqrt(2))/(4sqrt(2))=(1)/(2)" "`[from (3) and (6)]
Since `" "(AB)/(PQ)=(BC)/(QR)=(AC)/(PR)" "` `(` each equals `(1)/(2) )`
` therefore" "DeltaABC ~DeltaPQR" "(because" corresponding sides are proportional")`