Question

1. in figure 5, rotate quadrilateral abcd 60° counter - clockwise using center b.
2. in figure 6, rotate quadrilateral abcd 60° clockwise using center c.
3. in figure 7, reflect quadrilateral abcd using line ℓ.
4. in figure 8, translate quadrilateral abcd so that a goes to c.

Explanation:

Step1: Recall rotation rules

For a rotation of a point $(x,y)$ counter - clockwise about a center $(a,b)$ by an angle $\theta$, the transformation formula is $x'=(x - a)\cos\theta-(y - b)\sin\theta+a$ and $y'=(x - a)\sin\theta+(y - b)\cos\theta + b$. For a clock - wise rotation, we use $\theta=-\theta$ in the above formulas. For reflection over a line $y = mx + c$, we use the reflection formula. For translation, if a point $A(x_1,y_1)$ is translated to $C(x_2,y_2)$, the translation vector is $\vec{v}=(x_2 - x_1,y_2 - y_1)$ and we apply this vector to all points of the quadrilateral.

Step2: Solve problem 5

To rotate quadrilateral $ABCD$ 60° counter - clockwise about center $B$. Let the coordinates of $B$ be $(x_B,y_B)$ and of other points be $(x_i,y_i)$ for $i = A,C,D$. Using the rotation formula $x_i'=(x_i - x_B)\cos60^{\circ}-(y_i - y_B)\sin60^{\circ}+x_B$ and $y_i'=(x_i - x_B)\sin60^{\circ}+(y_i - y_B)\cos60^{\circ}+y_B$. Plot the new points $A',B,C',D'$ to get the rotated quadrilateral.

Step3: Solve problem 6

To rotate quadrilateral $ABCD$ 60° clockwise about center $C$. Let the coordinates of $C$ be $(x_C,y_C)$ and of other points be $(x_j,y_j)$ for $j=A,B,D$. Since it is a clock - wise rotation, we use $\theta=- 60^{\circ}$ in the rotation formula $x_j'=(x_j - x_C)\cos(-60^{\circ})-(y_j - y_C)\sin(-60^{\circ})+x_C$ and $y_j'=(x_j - x_C)\sin(-60^{\circ})+(y_j - y_C)\cos(-60^{\circ})+y_C$. Plot the new points $A',B',C,D'$ to get the rotated quadrilateral.

Step4: Solve problem 7

To reflect quadrilateral $ABCD$ over line $\ell$. First, find the equation of line $\ell$ if not given. For each point $(x_k,y_k)$ of the quadrilateral ($k = A,B,C,D$), use the reflection formula. If the line is $y=mx + c$, the distance from the point $(x_k,y_k)$ to the line is $d=\frac{|y_k-mx_k - c|}{\sqrt{1 + m^{2}}}$. The reflected point $(x_k',y_k')$ can be found using the properties of perpendicular distance from the point to the line and mid - point on the perpendicular line segment between the point and its reflection. Plot the new points $A',B',C',D'$ to get the reflected quadrilateral.

Step5: Solve problem 8

To translate quadrilateral $ABCD$ so that $A$ goes to $C$. First, find the translation vector $\vec{v}=(x_C - x_A,y_C - y_A)$ where $(x_A,y_A)$ are coordinates of $A$ and $(x_C,y_C)$ are coordinates of $C$. Then for each point $(x_n,y_n)$ of the quadrilateral ($n = A,B,C,D$), the new point $(x_n',y_n')=(x_n+(x_C - x_A),y_n+(y_C - y_A))$. Plot the new points $A',B',C',D'$ to get the translated quadrilateral.

Since we are not given the actual coordinates of the points and the equation of the line $\ell$, we can only provide the general steps.

Answer:

The new quadrilaterals for each transformation can be obtained by following the above - described steps for rotation, reflection and translation.