Question

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

Explanation:

Step1: Identify transformation rule

The rule is a \(90^{\circ}\) rotation about origin then \(y\) - axis reflection.

Step2: Recall rotation formula

For \(90^{\circ}\) rotation about origin \((x,y)\to(-y,x)\).

Step3: Recall reflection formula

For \(y\) - axis reflection \((x,y)\to(-x,y)\).

Step4: Combine transformation rules

Combined rule is \((x,y)\to(y,x)\).

Step5: Apply to vertices

Apply to \(\triangle ABC\) vertices and compare with given triangles.

Answer:

1. First, understand the transformation rule:

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

1. Analyze the vertices of \(\triangle ABC\):

• Let's assume the vertices of \(\triangle ABC\) are \(A(x_1,y_1)\), \(B(x_2,y_2)\), and \(C(x_3,y_3)\). After the transformation \(r_{y - axis}\circ R_{O,90^{\circ}}\), the vertices will be \(A'(y_1,x_1)\), \(B'(y_2,x_2)\), and \(C'(y_3,x_3)\).
• By applying these transformation rules to the vertices of \(\triangle ABC\) and comparing with the given triangles on the grid:
• When we perform the \(90^{\circ}\) rotation about the origin on \(\triangle ABC\) and then the \(y\) - axis reflection, we find that the final image is triangle 1.