Question

which choices correctly describe reflections in the diagram? choose three correct answers.
the triangle a is reflected across the x - axis onto triangle b.
the triangle a is reflected across the y - axis onto triangle d.
the triangle d is reflected across the line y = x onto triangle b.
the triangle d is reflected across the x - axis onto triangle c.

Explanation:

Step1: Analyze reflection of triangle A to B

For a reflection across the \(x -\)axis, the rule is \((x,y)\to(x, - y)\). If we check the vertices of triangle \(A\) and \(B\), we can see that the \(x -\)coordinates remain the same and the \(y -\)coordinates change their signs. So, triangle \(A\) is reflected across the \(x -\)axis onto triangle \(B\).

Step2: Analyze reflection of triangle D to C

For a reflection across the \(y -\)axis, the rule is \((x,y)\to(-x,y)\). If we check the vertices of triangle \(D\) and \(C\), we can see that the \(y -\)coordinates remain the same and the \(x -\)coordinates change their signs. So, triangle \(D\) is reflected across the \(y -\)axis onto triangle \(C\).

Step3: Analyze reflection of triangle D to B (using line \(y = x\))

For a reflection across the line \(y=x\), the rule is \((x,y)\to(y,x)\). Let's assume a vertex of triangle \(D\) is \((a,b)\), after reflection across \(y = x\) it becomes \((b,a)\). If we check the vertices of triangle \(D\) and \(B\) (by looking at their positions on the coordinate - plane), we can verify the reflection rule.

Answer:

The triangle A is reflected across the \(x -\)axis onto triangle B; The triangle D is reflected across the \(y -\)axis onto triangle C; The triangle D is reflected across the line \(y = x\) onto triangle B.