Question

triangle mno is the image of triangle mno under a translation left 3 units and down 6 units followed by a rotation about the origin. write the rules for the translation and rotation. translation: (x, y) → ( , ) rotation: (x, y) → ( , )

Explanation:

Step1: Determine translation rule

A translation left 3 units and down 6 units changes the \(x\) - coordinate by subtracting 3 and the \(y\) - coordinate by subtracting 6. The rule for translation is \((x,y)\to(x - 3,y - 6)\).

Step2: Determine rotation rule

To find the rotation rule, we can use a point - by - point analysis. Let's assume a general point \((x,y)\) and observe the transformation from the pre - image to the image after translation and rotation. If we consider the relationship between the pre - image and image points, we can see that a \(90^{\circ}\) clockwise rotation about the origin has the rule \((x,y)\to(y,-x)\). We can verify this by looking at the coordinates of the vertices of the triangle before and after the combined transformation.

Answer:

Translation: \((x,y)\to(x - 3,y - 6)\)
Rotation: \((x,y)\to(y,-x)\)