Question

the rule $r_{y - axis} \circ r_{0,90\circ}(x,y)$ is applied to $\triangle abc$. which triangle shows the final image? 1 2 3 4

Explanation:

Step1: Identify the transformation rules

The composition \(r_{y - axis}\circ R_{0,90^{\circ}}\) first rotates \(90^{\circ}\) counter - clockwise about the origin (\(R_{0,90^{\circ}}(x,y)=(-y,x)\)) and then reflects across the \(y\) - axis (\(r_{y - axis}(x,y)=(-x,y)\)). The combined rule for a point \((x,y)\) is \((y,x)\).

Step2: Apply the transformation to \(\triangle ABC\)

For each vertex of \(\triangle ABC\), apply the combined transformation rule \((y,x)\) to get the new vertices of the transformed triangle.

Step3: Match with the given triangles

By comparing the orientation and position of the vertices of the transformed triangle with the given triangles 1, 2, 3, and 4, we find that triangle 1 is the final image of \(\triangle ABC\) after the transformation \(r_{y - axis}\circ R_{0,90^{\circ}}\).

So the answer is 1.

Answer:

1. First, understand the transformation rules:

• The rule \(r_{y - axis}\circ R_{0,90^{\circ}}(x,y)\) means a composition of two transformations. First, we perform a \(90 - degree\) counter - clockwise rotation about the origin \(R_{0,90^{\circ}}(x,y)=(-y,x)\).
• Then, we perform a reflection across the \(y\) - axis. The rule for reflection across the \(y\) - axis is \(r_{y - axis}(x,y)=(-x,y)\).
• Combining the two, if we start with a point \((x,y)\), after the rotation \(R_{0,90^{\circ}}\), we get \((-y,x)\), and after the reflection \(r_{y - axis}\), we get \((y,x)\).

1. Analyze the original triangle \(\triangle ABC\):

• Let's assume the vertices of \(\triangle ABC\) are \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\). After the transformation \(r_{y - axis}\circ R_{0,90^{\circ}}\), the new vertices will be \((y_1,x_1)\), \((y_2,x_2)\), \((y_3,x_3)\).
• By visual inspection of the given triangles on the grid:
• When we perform the \(90 - degree\) counter - clockwise rotation about the origin on \(\triangle ABC\) and then reflect across the \(y\) - axis, we can match the orientation and position of the vertices of the transformed triangle with one of the given triangles.
• The triangle that matches the final image of \(\triangle ABC\) after the transformation \(r_{y - axis}\circ R_{0,90^{\circ}}\) is triangle 1.