Question

2 trapezoid pqrs and trapezoid tuvw are graphed on the coordinate grid shown below. which series of two transformations can be used to map trapezoid pqrs onto trapezoid tuvw? a rotation of ( 180^{circ} ) centered at ( (4,4) ) a dilation with a scale factor of 2 centered at ( (5,4) ) a rotation of ( 180^{circ} ) centered at ( (4,4) ) a dilation with a scale factor of 2 centered at the origin a rotation of ( 180^{circ} ) centered at the origin a dilation with a scale factor of 2 centered at ( (5,4) ) a rotation of ( 180^{circ} ) centered at the origin a dilation with a scale factor of 2 centered at the origin

Explanation:

Step1: Analyze the rotation

First, consider the rotation. If we rotate trapezoid \(PQRS\) \(180^{\circ}\) centered at \((4,4)\). The formula for a \(180^{\circ}\) rotation about a point \((a,b)\) is \((x,y)\to(2a - x,2b - y)\).
Let's take a point from \(PQRS\), say \(P(3,1)\). After rotation about \((4,4)\), it becomes \((2\times4 - 3,2\times4 - 1)=(5,7)\).

Step2: Analyze the dilation

After rotation, we need to check the dilation. A dilation with a scale factor \(k = 2\) centered at \((5,4)\). The formula for dilation about a point \((a,b)\) is \((x,y)\to(k(x - a)+a,k(y - b)+b)\).
Take the rotated point \((5,7)\). After dilation: \((2(5 - 5)+5,2(7 - 4)+4)=(5,10)\) (matches the top - right point of \(TUVW\) after considering the trapezoid structure).
If we consider rotation about the origin \((0,0)\) (formula \((x,y)\to(-x,-y)\) for \(180^{\circ}\) rotation), and dilation about the origin (formula \((x,y)\to(kx,ky)\)), the position of the trapezoid \(TUVW\) relative to \(PQRS\) does not match the coordinate - grid based on the center of dilation and rotation. For example, if we rotate \(P(3,1)\) about the origin \(180^{\circ}\) we get \((- 3,-1)\), and then dilating about the origin with scale factor \(2\) gives \((-6,-2)\) which is not in the correct position relative to \(TUVW\).

Answer:

A. a rotation of \(180^{\circ}\) centered at \((4,4)\), a dilation with a scale factor of \(2\) centered at \((5,4)\)