Question

Can someone help me map these two triangles?

Answer 1

`(AB) / (XY) = (BC) / (YZ) = (CA) / (FD) = 1/2`

Hence the two triangles are similar.

Explanation:

In triangle ABC : coordinates of A(0,0), B(1,3), C(4,2)

In triangle XYZ : coordinates of X(-5,1), Y ( -3,-5), Z(3,-3)

Distance between two points is calculated using the formula

`d = sqrt((x2-x1)^2 + (y2-y1)^2)`

`vec(AB) = sqrt((1-0)^2 + (3-0)^2 = sqrt10`

`vec(BC) = sqrt((4-1)^2 + (2-3)^2) = sqrt10`

`vec(CA) = sqrt((4-0)^2 + (2-0)^2) = sqrt20`

`vec(DE) = sqrt((-5+3)^2 + (1+5)^2 = sqrt40 = 2sqrt10`

`vec(EF) = sqrt((3+3)^2 + (-3+5)^2) = sqrt40 = 2sqrt10`

`vec(FD) = sqrt((3+5)^2 + (-3-1)^2) = sqrt80 = 2sqrt20`

For two triangles are to be similar, their sides are to be in the same proportion.

i.e. `(AB) / (XY) = (BC) / (YZ) = (CA) / (FD)`

Let us check whether this condition is satisfied in the given sum.

`cancel(sqrt10)^color(red)1 / (2 cancelsqrt10) = cancel(sqrt10)^color(red)1 / (2 cancelsqrt10) = cancel(sqrt20)^color(red)1 / (2cancelsqrt20)`

`=>. 1/2 = 1/2 = 1/2`

Hence the two triangles are similar.