Question

1. in figure 5, rotate quadrilateral abcd 60° counterclockwise using center b.

Explanation:

Step1: Recall rotation rules

To rotate a point $P(x,y)$ counter - clockwise about a center $O(a,b)$ by an angle $\theta$, we use the rotation matrix. In a grid - based approach, we can use geometric properties. For a $60^{\circ}$ counter - clockwise rotation about a point $B$, we consider the vectors from $B$ to other points of the quadrilateral.

Step2: Rotate point $A$

Measure the distance and angle of the vector $\overrightarrow{BA}$. Then, rotate this vector $60^{\circ}$ counter - clockwise about $B$. Using the properties of equilateral triangles (since a $60^{\circ}$ rotation in a plane can be related to equilateral triangle construction), we find the new position of $A$ relative to $B$.

Step3: Rotate point $C$

Similarly, find the vector $\overrightarrow{BC}$, rotate it $60^{\circ}$ counter - clockwise about $B$. We can use the fact that in a $60^{\circ}$ rotation, if we consider the right - angled triangles formed by the grid lines and the vectors, we can calculate the new coordinates of $C$ relative to $B$ by using trigonometric relations (for example, if the side lengths of the vectors in terms of grid units are known, we can use $\cos60^{\circ}=\frac{1}{2}$ and $\sin60^{\circ}=\frac{\sqrt{3}}{2}$).

Step4: Rotate point $D$

Find the vector $\overrightarrow{BD}$, rotate it $60^{\circ}$ counter - clockwise about $B$ to get the new position of $D$.

Step5: Connect the new points

Connect the new positions of $A$, $C$, and $D$ to $B$ and to each other to form the rotated quadrilateral.

Answer:

The new quadrilateral is formed by the rotated positions of $A$, $C$, and $D$ about point $B$ by $60^{\circ}$ counter - clockwise, which can be constructed geometrically on the grid as described above.