Question

the graph below shows the transformation from triangle 1 to triangle 2.
which sequence of steps would transform triangle 1 to triangle 2?

• reflect across the y - axis; rotate 180° counterclockwise about the origin
• reflect across the x - axis; rotate 270° counterclockwise about the origin
• reflect across the y - axis; rotate 90° counterclockwise about the origin
• rotate 90° counterclockwise about the origin; rotate 270° counterclockwise about the origin

Explanation:

Step1: Analyze reflection and rotation properties

• Reflection across the \(y\) - axis changes the \(x\) - coordinate sign (\((x,y)\to(-x,y)\)).
• Rotation \(90^{\circ}\) counter - clockwise about the origin changes \((x,y)\to(-y,x)\).

Let's assume a point \((x,y)\) on triangle 1.

Step2: Check each option

• Option 1: Reflect across \(y\) - axis \((x,y)\to(-x,y)\), then rotate \(180^{\circ}\) counter - clockwise \((-x,y)\to(x, - y)\) (not correct).
• Option 2: Reflect across \(x\) - axis \((x,y)\to(x,-y)\), rotate \(270^{\circ}\) counter - clockwise \((x,-y)\to(y,x)\) (not correct).
• Option 3: Reflect across \(y\) - axis \((x,y)\to(-x,y)\), rotate \(90^{\circ}\) counter - clockwise \((-x,y)\to(-y,-x)\) (correct transformation).
• Option 4: Rotate \(90^{\circ}\) counter - clockwise \((x,y)\to(-y,x)\), rotate \(270^{\circ}\) counter - clockwise \((-y,x)\to(y,-x)\) (equivalent to no net rotation, not correct).

Answer:

reflect across the \(y\) - axis, rotate \(90^{\circ}\) counterclockwise about the origin.