Question

structure \\( \overline { x y } \\) has endpoints \\( x ( - 5,6 ) \\) and \\( y ( 0,4 ) \\), the image of \\( \overline { x y } \\) has the endpoints \\( x ^ { \prime } ( 6,5 ) \\) and \\( y ^ { \prime } ( 4,0 ) \\), and \\( \overline { x y } \cong \overline { x ^ { \prime } y ^ { \prime } } \\). identify the transformation.
rotation of about the

Explanation:

Step1: Recall the rotation rules

The rule for a rotation of \(180^{\circ}\) about the origin \((x,y)\to(-x,-y)\). The rule for a rotation of \(270^{\circ}\) counter - clockwise about the origin \((x,y)\to(y, - x)\). The rule for a rotation of \(360^{\circ}\) about the origin \((x,y)\to(x,y)\).

Step2: Check the transformation of the endpoints

For point \(X(-5,6)\):
If we use the rotation rule of \(270^{\circ}\) counter - clockwise about the origin \((x,y)\to(y,-x)\), when \(x = - 5\) and \(y = 6\), we get \((6,5)\) (since \(y = 6\) and \(-x=-(-5) = 5\)).
For point \(Y(0,4)\):
When \(x = 0\) and \(y = 4\), using the rule \((x,y)\to(y,-x)\), we get \((4,0)\) (since \(y = 4\) and \(-x=0\)).

Answer:

rotation of \(270^{\circ}\) about the origin.