Question

a triangle with points a(3, 4), b(6, 4), and
c(4, 3) is reflected over x - axis to form
iriangle abc. which of these
sequences of transformations will also
carry triangle abc into abc? select all
that apply.
a reflection across the y - axis followed by a reflection across the x - axis
b. a reflection across the y - axis followed by a reflection across the line with equation y = - x
c reflection across the y - axis followed by a 180° rotation around the origin
d reflection across the line with equation y = x followed by a 180° rotation around the origin

Explanation:

Step1: Recall reflection and rotation rules

• Reflection across the \(y -\)axis: \((x,y)\to(-x,y)\)
• Reflection across the \(x -\)axis: \((x,y)\to(x, - y)\)
• Rotation \(180^{\circ}\) around the origin: \((x,y)\to(-x,-y)\)
• Reflection across the line \(y = x\): \((x,y)\to(y,x)\)
• Reflection across the line \(y=-x\): \((x,y)\to(-y,-x)\)

For option A:

• First, reflect \(\triangle ABC\) across the \(y -\)axis. If \(A(3,4)\), \(B(6,4)\), \(C(4,3)\), after reflection across the \(y -\)axis, the points become \(A'(- 3,4)\), \(B'(-6,4)\), \(C'(-4,3)\)
• Then, reflect across the \(x -\)axis. The points become \(A''(-3,-4)\), \(B''(-6,-4)\), \(C''(-4,-3)\)

For option B:

• First, reflect \(\triangle ABC\) across the \(y -\)axis. Points \(A(3,4)\to A'(-3,4)\), \(B(6,4)\to B'(-6,4)\), \(C(4,3)\to C'(-4,3)\)
• Then, reflect across the line \(y=-x\). Using the rule \((x,y)\to(-y,-x)\), \(A'(-3,4)\to A''(-4,3)\), \(B'(-6,4)\to B''(-4,6)\), \(C'(-4,3)\to C''(-3,4)\)

For option C:

• Reflect \(\triangle ABC\) across the \(y -\)axis. \(A(3,4)\to A'(-3,4)\), \(B(6,4)\to B'(-6,4)\), \(C(4,3)\to C'(-4,3)\)
• Then, rotate \(180^{\circ}\) around the origin. Using the rule \((x,y)\to(-x,-y)\), \(A'(-3,4)\to A''(3,-4)\), \(B'(-6,4)\to B''(6,-4)\), \(C'(-4,3)\to C''(4,-3)\)

For option D:

• Reflect across the line \(y = x\). \(A(3,4)\to A'(4,3)\), \(B(6,4)\to B'(4,6)\), \(C(4,3)\to C'(3,4)\)
• Then, rotate \(180^{\circ}\) around the origin. Using the rule \((x,y)\to(-x,-y)\), \(A'(4,3)\to A''(-4,-3)\), \(B'(4,6)\to B''(-4,-6)\), \(C'(3,4)\to C''(-3,-4)\)

Answer:

B. reflection across the \(y -\)axis followed by a reflection across the line \(y=-x\)