Question

on the set of axes below, \\( \triangle let \\) and \\( \triangle let \\) are graphed in the coordinate plane where \\( \triangle let \cong \triangle let \\).
which sequence of rigid motions maps \\( \triangle let \\) onto \\( \triangle let \\)?

1. a reflection over the \\( y \\)-axis followed by a reflection over the \\( x \\)-axis
2. a rotation of \\( 180^{\circ} \\) about the origin
3. a rotation of \\( 90^{\circ} \\) counterclockwise about the origin followed by a reflection over the \\( y \\)-axis
4. a reflection over the \\( x \\)-axis followed by a rotation of \\( 90^{\circ} \\) clockwise about the origin

Explanation:

Step1: Analyze each option

• Option 1:
• Let \((x,y)\) be a point in \(\triangle LET\). A reflection over the \(y -\)axis changes the point to \((-x,y)\), and then a reflection over the \(x -\)axis changes it to \((-x,-y)\).
• Option 2:
• A rotation of \(180^{\circ}\) about the origin changes a point \((x,y)\) to \((-x,-y)\).
• Option 3:
• A rotation of \(90^{\circ}\) counter - clockwise about the origin changes a point \((x,y)\) to \((-y,x)\), and then a reflection over the \(y -\)axis changes it to \((y,x)\).
• Option 4:
• A reflection over the \(x -\)axis changes a point \((x,y)\) to \((x,-y)\), and then a rotation of \(90^{\circ}\) clockwise about the origin changes it to \((-y,x)\).

Step2: Check the transformation

• Rigid motions preserve the shape and size of the figure. By observing the position of \(\triangle LET\) and \(\triangle L''E''T''\) in the coordinate plane, if we consider a general point \((x,y)\) in \(\triangle LET\) and apply a rotation of \(180^{\circ}\) about the origin \((x,y)\to(-x,-y)\) which maps \(\triangle LET\) onto \(\triangle L''E''T''\)

Answer:

1. a rotation of \(180^{\circ}\) about the origin