Question

1. a rectangle has vertices at (2, 5), (6, 5), (6, 3), and (2, 3). which one transformation will result in the figure mapping onto itself? a rotation of 90° about the point (4,4).

Explanation:

Step1: Recall rotation properties

A rectangle maps onto itself after a 180 - degree rotation about its center. The center of a rectangle with vertices \((x_1,y_1),(x_2,y_1),(x_2,y_2),(x_1,y_2)\) is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). For the rectangle with vertices \((2,5),(6,5),(6,3),(2,3)\), the center is \((\frac{2 + 6}{2},\frac{5+3}{2})=(4,4)\). A 90 - degree rotation about \((4,4)\) will not map the rectangle onto itself. A 180 - degree rotation about \((4,4)\) will map it onto itself. But among the given options, we analyze the 90 - degree rotation.
A 90 - degree rotation of a point \((x,y)\) about a center \((a,b)\) is given by the transformation \((x,y)\to(a+(y - b),b-(x - a))\). Let's take a vertex, say \((2,5)\) and rotate it 90 - degree about \((4,4)\).

Step2: Apply rotation formula

For a point \((x,y)=(2,5)\) and center \((a,b)=(4,4)\), the new point \((x',y')\) after 90 - degree rotation is \(x'=4+(5 - 4)=5\) and \(y'=4-(2 - 4)=6\). This new point is not a vertex of the original rectangle.
Since a 90 - degree rotation about \((4,4)\) does not map the rectangle onto itself, none of the given transformation (as only 90 - degree rotation about \((4,4)\) is given) will map the rectangle onto itself. But if we assume we are looking for a correct transformation, a 180 - degree rotation about \((4,4)\) would map the rectangle onto itself. However, based on the options provided, the answer is that the given 90 - degree rotation will not map the rectangle onto itself.

Answer:

The given 90 - degree rotation about the point \((4,4)\) will not result in the figure mapping onto itself.