Please Consider the function f : R -> R given by f(x, y) = (2 -...

Please Consider the function f : R -> R given by f(x, y) = (2 - y, 2 - x).

(a) Prove that f is an isometry.

(b) Draw the triangle with vertices A = (1, 2), B = (3, 1), C = (3, 2), and the triangle with vertices f(A), f(B), f(C).

Solutions

Expert Solution

(a) Lets prove that the function f preserve the distance. For this. let us assume any two points P (x₁, y₁) and another point Q (x₂, y₂), then

Using the definition of isometry f : R --> R, we have

Now, using the distance formula for points P' and Q', we have

Therefore, f is an isometry.

(b) The triangle ABC, with vertices A = (1, 2), B = (3, 1), C = (3, 2), and the triangle A'B'C' with vertices f(A)=A', f(B)=B', and f(C)=C' are shown in figure below:

(c) We see from the figure that the triangle ABC is the reflection about the line x + y = 2. So, the triangle A'B'C' can be obtained by roatating clockwise direction (270^o - 2 tan^-1(-1/2)) about point (-1, 3). So, it requies both translation and rotation. So, we can put it under a glide reflection

milcah answered 1 year ago

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