Question

which are true if figure defg is reflected across the x-axis? check all that apply. d(0, 4) → d(0, −4) e(−2, 0) → e(−2, 0) the perpendicular distance from g to the x-axis will equal 2 units. the perpendicular distance from d to the x-axis will equal 8 units. the orientation will be preserved.

Explanation:

Step1: Recall reflection over x - axis rule

The rule for reflecting a point \((x,y)\) across the \(x\) - axis is \((x,y)\to(x, - y)\).

Step2: Analyze point D

For point \(D(0,4)\), applying the reflection rule across the \(x\) - axis, we get \(D'(0,-4)\). So the statement \(D(0,4)\to D'(0, - 4)\) is true.

Step3: Analyze point E

For point \(E(-2,0)\), applying the reflection rule \((x,y)\to(x, - y)\), we have \(E'(-2,0)\) (since \(y = 0\), \(-y=0\)). So the statement \(E(-2,0)\to E'(-2,0)\) is true.

Step4: Analyze point G

First, we need to find the coordinates of \(G\). From the graph, we can see that \(G\) has coordinates \((4,2)\). When we reflect \(G\) across the \(x\) - axis, the new point \(G'\) will have coordinates \((4,-2)\). The perpendicular distance from a point \((x,y)\) to the \(x\) - axis is \(|y|\). For \(G'\), \(y=-2\), so the perpendicular distance is \(|-2| = 2\) units. So the statement "The perpendicular distance from \(G'\) to the \(x\) - axis will equal 2 units" is true.

Step5: Analyze point D'

For point \(D'(0,-4)\), the perpendicular distance from \(D'\) to the \(x\) - axis is \(|-4|=4\) units, not 8 units. So the statement "The perpendicular distance from \(D'\) to the \(x\) - axis will equal 8 units" is false.

Step6: Analyze orientation

Reflection across the \(x\) - axis changes the orientation (it is a flip over the \(x\) - axis, so the orientation is reversed). So the statement "The orientation will be preserved" is false.

Answer:

• \(D(0, 4)\to D'(0, - 4)\)
• \(E(-2, 0)\to E'(-2, 0)\)
• The perpendicular distance from \(G'\) to the \(x\) - axis will equal 2 units.