Question

1. translate quadrilateral abcd so that point a, which corresponds to point a, is located at coordinates (-4,3). label the other vertices b to correspond to b, c to c, and d to d. what are the coordinates of a, b, c, and d?

a = ____ b = __ c = __ d = ____

1. reflect quadrilateral abcd across line m. label the coordinates a, b, c, and d so that point a corresponds to the reflection of point a, b corresponds to the reflection of b, and c corresponds to the reflection of c. what are the coordinates of a, b, c, and d?

a = ____ b = __ c = __ d = ____

1. rotate quadrilateral abcd 90° counterclockwise around point d. label the points a, b, c, and d so that a corresponds to the rotation of point a, b corresponds to b, c to c, and d to d. what are the coordinates of a, b, c, and d?

a = ____ b = __ c = __ d = ____
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Explanation:

Step1: Determine original coordinates

Assume from the grid, \(A=(1,1)\), \(B=(1,3)\), \(C=(3,3)\), \(D=(4,1)\)

Step2: Calculate translation (for part 1)

To get from \(A=(1,1)\) to \(A'=(- 4,3)\), the translation rule is \((x,y)\to(x - 5,y + 2)\).

• \(B'=(1-5,3 + 2)=(-4,5)\)
• \(C'=(3-5,3 + 2)=(-2,5)\)
• \(D'=(4-5,1 + 2)=(-1,3)\)

Step3: Reflection rules (for part 2, assume line \(m\) is not given, assume reflection over \(y - axis\): \((x,y)\to(-x,y)\))

• \(A''=(-1,1)\)
• \(B''=(-1,3)\)
• \(C''=(-3,3)\)
• \(D''=(-4,1)\)

Step4: Rotation rules (for part 3)

The rule for a \(90^{\circ}\) counter - clockwise rotation around a point \((a,b)\) is \((x,y)\to(a-(y - b),b+(x - a))\). Here \(a = 4\), \(b = 1\).

• For \(A\): \(x = 1\), \(y = 1\), \(A'''=4-(1 - 1),1+(1 - 4)=(4,-2)\)
• For \(B\): \(x = 1\), \(y = 3\), \(B'''=4-(3 - 1),1+(1 - 4)=(2,-2)\)
• For \(C\): \(x = 3\), \(y = 3\), \(C'''=4-(3 - 1),1+(3 - 4)=(2,0)\)
• \(D'''=(4,1)\)

Answer:

1. \(A'=(-4,3)\), \(B'=(-4,5)\), \(C'=(-2,5)\), \(D'=(-1,3)\)
2. \(A''=(-1,1)\), \(B''=(-1,3)\), \(C''=(-3,3)\), \(D''=(-4,1)\)
3. \(A'''=(4,-2)\), \(B'''=(2,-2)\), \(C'''=(2,0)\), \(D'''=(4,1)\)