Question

1. rhombus math is graphed below. which of the following will carry the rhombus onto itself? 1.) a reflection over the line y = 0 2.) a reflection over the line y = x 3.) a reflection over the line y = 2 4.) a reflection over the line y = -2x + 6

Explanation:

Step1: Check reflection over \(y = 0\)

A reflection over \(y = 0\) (the \(x -\)axis) changes the \(y -\)coordinate of a point \((x,y)\) to \((x, - y)\). For example, if a point \(P(x,y)\) is on the rhombus, its image \(P'(x,-y)\) after reflection over \(y = 0\) will not map the rhombus onto itself.

Step2: Check reflection over \(y=x\)

A reflection over \(y = x\) swaps the \(x\) and \(y\) coordinates of a point \((x,y)\) to \((y,x)\). This will not map the rhombus onto itself.

Step3: Check reflection over \(y = 2\)

Let's assume a point \((x,y)\) on the rhombus. The formula for reflection over the line \(y = c\) is \((x,2c - y)\). Here \(c = 2\), so the reflection of a point \((x,y)\) is \((x,4 - y)\).
If we consider the vertical distance of points from the line \(y=2\). The rhombus is symmetric about the line \(y = 2\). For any point \((x,y)\) on the rhombus, its reflection \((x,4 - y)\) is also on the rhombus.

Step4: Check reflection over \(y=-2x + 6\)

The general formula for reflection over a line \(Ax+By + C=0\) (rewrite \(y=-2x + 6\) as \(2x+y-6 = 0\)) is complex. Using the property of symmetry of the rhombus (by inspection of the graph), the rhombus is not symmetric about the line \(y=-2x + 6\).

Answer:

1. a reflection over the line \(y = 2\)