Question

which pair of rigid transformations will map triangle abc onto abc?
a. a reflection across x - axis and then reflection across y - axis.
b. a reflection across line y = x,
then reflection y = -x
c. a rotation 90 degree counterclockwise around the origin, then reflection across x - axis
d. a rotation 90 degrees clockwise around origin, then a rotation 90 degrees counterclockwise around origin

Explanation:

Step1: Analyze each option

• Option A: A reflection across the \(x -\)axis changes the \(y -\)coordinate sign (\((x,y)\to(x, - y)\)), then a reflection across the \(y -\)axis changes the \(x -\)coordinate sign (\((x,-y)\to(-x,-y)\)).
• Option B: A reflection across the line \(y = x\) swaps \(x\) and \(y\) coordinates (\((x,y)\to(y,x)\)), then a reflection \(y=-x\) swaps and changes signs (\((y,x)\to(-x, - y)\)).
• Option C: A \(90 -\)degree counterclockwise rotation around the origin has the transformation rule \((x,y)\to(-y,x)\), then a reflection across the \(x -\)axis changes the \(y -\)coordinate sign (\((-y,x)\to(-y,-x)\)).
• Option D: A \(90 -\)degree rotation \(90\) degrees clockwise around the origin has the rule \((x,y)\to(y, - x)\), then a \(90 -\)degree rotation around the origin counterclockwise (\((y,-x)\to(x,y)\)).

Step2: Check the transformation of points

Let's assume a general point \((x,y)\) on \(\triangle EFG\).
For a \(90 -\)degree clockwise rotation around the origin \((x,y)\to(y, - x)\), then a \(90 -\)degree counterclockwise rotation around the origin \((y,-x)\to(x,y)\) (equivalent to a reflection).

Answer:

D. A rotation \(90\) degrees clockwise around origin, then a rotation \(90\) degrees counterclockwise around origin.