Question

quadrilateral abcd drawn on the coordinate plane is shown below.
first, abcd is rotated 90° counterclockwise around the origin.
second, the image of the rotation is translated 2 units to the left, 1 unit up.
third, the image of the translation is dilated with a scale factor of 2 with center at the origin.
plot the points of the quadrilateral after the sequence of transformations.
graphing

Explanation:

Step1: Find the coordinates of \(ABCD\)

From the graph, \(A(1,4)\), \(B(-1,2)\), \(C(2,-1)\), \(D(4,1)\)

Step2: Rotate \(90^{\circ}\) counter - clockwise around the origin

The rule for a \(90^{\circ}\) counter - clockwise rotation about the origin \((x,y)\to(-y,x)\)

• For \(A(1,4)\): \((1,4)\to(-4,1)\)
• For \(B(-1,2)\): \((-1,2)\to(-2,-1)\)
• For \(C(2,-1)\): \((2,-1)\to(1,2)\)
• For \(D(4,1)\): \((4,1)\to(-1,4)\)

Step3: Translate 2 units to the left and 1 unit up

The rule for translation \(2\) units left and \(1\) unit up is \((x,y)\to(x - 2,y + 1)\)

• For \((-4,1)\): \((-4-2,1 + 1)=(-6,2)\)
• For \((-2,-1)\): \((-2-2,-1 + 1)=(-4,0)\)
• For \((1,2)\): \((1-2,2 + 1)=(-1,3)\)
• For \((-1,4)\): \((-1-2,4 + 1)=(-3,5)\)

Step4: Dilate with a scale factor of 2

The rule for dilation with a scale factor \(k = 2\) about the origin is \((x,y)\to(2x,2y)\)

• For \((-6,2)\): \((-6\times2,2\times2)=(-12,4)\)
• For \((-4,0)\): \((-4\times2,0\times2)=(-8,0)\)
• For \((-1,3)\): \((-1\times2,3\times2)=(-2,6)\)
• For \((-3,5)\): \((-3\times2,5\times2)=(-6,10)\)

Answer:

The coordinates of the vertices of the transformed quadrilateral are \(A'(-12,4)\), \(B'(-8,0)\), \(C'(-2,6)\), \(D'(-6,10)\)