Question

pezoid s’t’u’v’ is created by reflecting trapezoid stuv over a line. the coordinates of the vertices of the pezoids are shown in the table.
trapezoid stuv | trapezoid stuv
(2, 2) | (2, −6)
(3, 6) | (3, −10)
(6, 7) | (6, −11)
(8, 4) | (8, −8)
complete the sentences to create true statements about the trapezoids.
trapezoid stuv is reflected over the line blank.
the rule (x, y)→( blank, blank ) can be used to create trapezoid s’t’u’v’.

Explanation:

Step1: Analyze x - coordinates

For each vertex, the x - coordinate of \(S'T'U'V'\) is the same as that of \(STUV\) (e.g., \(S(2,2)\) and \(S'(2, - 6)\) have the same \(x = 2\)). So the reflection is vertical (over a horizontal line), and the \(x\) - coordinate remains unchanged. So the first part of the rule is \(x\) (i.e., the \(x\) - coordinate in the image is the same as the original).

Step2: Find the mid - line (horizontal line of reflection)

To find the horizontal line of reflection, we use the mid - point formula for the \(y\) - coordinates of a point and its image. For a point \((x,y)\) and its image \((x,y')\), the line of reflection \(y = k\) satisfies \(k=\frac{y + y'}{2}\).
Take the point \(S(2,2)\) and \(S'(2,-6)\): \(k=\frac{2+( - 6)}{2}=\frac{-4}{2}=-2\). Let's check with another point, say \(T(3,6)\) and \(T'(3,-10)\): \(k=\frac{6+( - 10)}{2}=\frac{-4}{2}=-2\). So the line of reflection is \(y=-2\).

Step3: Find the rule for \(y\) - coordinate

We know that for a reflection over \(y = k\), the formula for the image of \((x,y)\) is \((x,2k - y)\). Here \(k=-2\), so \(2k - y=2\times(-2)-y=-4 - y\). Let's verify with \(S(2,2)\): \(2\times(-2)-2=-4 - 2=-6\), which matches \(S'(2,-6)\). With \(T(3,6)\): \(2\times(-2)-6=-4 - 6=-10\), which matches \(T'(3,-10)\). So the rule is \((x,y)\to(x,2\times(-2)-y)=(x,-4 - y)\) or we can also think in terms of the distance from the line of reflection. The distance from \(y\) to \(y = - 2\) is \(|y+2|\), and the image is on the other side of \(y=-2\) at the same distance, so \(y'=-2-(y - (-2))=-2 - y - 2=-4 - y\).

Answer:

Trapezoid \(STUV\) is reflected over the line \(y = - 2\).
The rule \((x,y)\to(x,-4 - y)\) (or \((x,y)\to(x,2\times(-2)-y)\)) can be used to create trapezoid \(S'T'U'V'\).