Question

find a congruence transformation that maps \\( \triangle zbd \\) to \\( \triangle tfy \\).
which congruence transformation maps the preimage to the congruent image? select the correct choice below and fill in the answer boxes to complete you choice.
\\( \bigcirc \\) a. \\( t \\) \\( \circ r_{(270^{\circ}, o)}(\triangle zbd) \\)
\\( \bigcirc \\) b. \\( t \\) \\( \circ r_{x - \text{axis}}(\triangle zbd) \\)
\\( \bigcirc \\) c. \\( t \\) \\( \circ r_{y - \text{axis}}(\triangle zbd) \\)
\\( \bigcirc \\) d. \\( t \\) \\( \circ r_{(90^{\circ}, o)}(\triangle zbd) \\)

Explanation:

Step1: Analyze rotation

First, consider the rotation \(r_{(90^{\circ},O)}\). When a point \((x,y)\) is rotated \(90^{\circ}\) counter - clockwise about the origin \(O\), the transformation rule is \((x,y)\to(-y,x)\).

Step2: Analyze translation

After rotation, consider the translation \(T_{(90,0)}\). A translation \(T_{(a,b)}\) moves a point \((x,y)\) to \((x + a,y + b)\). Here \(a = 90\) (units in the \(x\) - direction) and \(b=0\) (no movement in the \(y\) - direction).

Since the other options (A: rotation \(270^{\circ}\) which is equivalent to \(- 90^{\circ}\) rotation; B: reflection over \(x\) - axis \(R_{x - axis}(x,y)=(x,-y)\) followed by a wrong translation; C: reflection over \(y\) - axis \(R_{y - axis}(x,y)=(-x,y)\) followed by an undefined translation) do not map \(\triangle ZBD\) to \(\triangle TFY\) correctly.

Answer:

D. \(T_{(90,0)}\circ r_{(90^{\circ},O)}(\triangle ZBD)\)