Question

1. in the graph below, what is the line of reflection for △xyz and △xyz? x=2 the y-axis the x-axis

Explanation:

Step1: Recall the property of reflection

When a figure is reflected over a vertical line \(x = a\), the distance of each point from the line \(x=a\) is the same on both sides of the line.

Step2: Analyze the given triangles

Let's take a pair of corresponding points. For example, assume a point \((x,y)\) in \(\triangle XYZ\) and its image \((x',y')\) in \(\triangle X'Y'Z'\). The formula for reflection over the line \(x = a\) is \(x'=2a - x\).
If we observe the horizontal distance of corresponding points from the line \(x = 2\). Let's say a point \(P(x_1,y_1)\) in \(\triangle XYZ\) and \(P'(x_2,y_2)\) in \(\triangle X'Y'Z'\). The mid - point of the line segment joining \(P\) and \(P'\) has an \(x\) - coordinate of \(\frac{x_1 + x_2}{2}=2\) (using the mid - point formula \((\frac{x_1+x_2}{2},\frac{y_1 + y_2}{2})\) for two points \((x_1,y_1)\) and \((x_2,y_2)\)).
If we check the \(x\) - coordinates of corresponding vertices of \(\triangle XYZ\) and \(\triangle X'Y'Z'\), we can verify that the line \(x = 2\) is the perpendicular bisector of the line segments joining corresponding vertices.

For reflection over the \(x\) - axis, the \(y\) - coordinate of a point \((x,y)\) changes to \((x, - y)\). For reflection over the \(y\) - axis, the \(x\) - coordinate of a point \((x,y)\) changes to \((-x,y)\). But in our case, the \(x\) - coordinates of corresponding points do not follow \(x\to - x\) (for \(y\) - axis reflection) or \(y\to - y\) (for \(x\) - axis reflection) rule.

Answer:

\(x = 2\)