Question

1. which of the following algebraic rules is equivalent to the rotation in #4? (a) (x,y)→(y,x) (b) (x,y)→(-x,y) (c) (x,y)→(-y,x) (d) (x,y)→(-x,-y) 6. find the image of c using the same rotation. plot it and finish drawing abc. reasoning 7. parallelograms are four - sided figures that have two pairs of opposite, parallel sides. quadrilateral abcd shown below is a parallelogram. one of its two diagonals, ac, is drawn and its mid - point m is located. use tracing paper in this problem to help you answer the following question. (a) if cd was rotated 180° about m, explain why it would have to lie on top of ab. (b) for the same reason, if ad was rotated 180° about m, what line would it lie on top of? (c) at what point would the images of cd and ad have to intersect? based on this, what can you conclude about opposite sides and opposite angles of a parallelogram? explain your answer.

Explanation:

5.

Step1: Recall rotation rule

A 180 - degree rotation about the origin has the rule $(x,y)\to(-x,-y)$.

6. (No specific details given about point C in the question - assuming we know the coordinates of C as $(x,y)$)

Step1: Apply 180 - degree rotation rule

If $C=(x,y)$, after a 180 - degree rotation about the origin, the image $C'=(-x,-y)$. Then plot the new point and draw $\triangle A'B'C'$.

7. (a)

Step1: Use properties of parallelogram

In parallelogram $ABCD$, $M$ is the mid - point of diagonal $AC$. Since $AB\parallel CD$ and $M$ is the mid - point of the diagonal connecting opposite vertices, a 180 - degree rotation of $\overline{CD}$ about $M$ will map it onto $\overline{AB}$ because a 180 - degree rotation about the mid - point of the diagonal of a parallelogram swaps the opposite sides due to the symmetry of the parallelogram about the mid - point of its diagonal.

7. (b)

Step1: Identify opposite side

In parallelogram $ABCD$, if $\overline{AD}$ is rotated 180 degrees about $M$, it will lie on top of $\overline{BC}$ because in a parallelogram, opposite sides are parallel and equal in length, and a 180 - degree rotation about the mid - point of the diagonal preserves these properties and swaps the opposite sides.

7. (c)

Step1: Consider intersection point

The images of $\overline{CD}$ and $\overline{AD}$ after 180 - degree rotation about $M$ will intersect at the image of point $D$ under the 180 - degree rotation about $M$. We can conclude that opposite sides of a parallelogram are equal in length and parallel, and opposite angles are equal. This is because the 180 - degree rotation about the mid - point of the diagonal shows the symmetry of the parallelogram, and corresponding parts of the rotated figure match up in a way that verifies these properties.

Answer:

1. The rule for a 180 - degree rotation about the origin is $(x,y)\to(-x,-y)$.
2. If $C=(x,y)$, $C'=(-x,-y)$. Plot $C'$ and draw $\triangle A'B'C'$.
3. (a) Because of the symmetry of the parallelogram about the mid - point of the diagonal and the parallel and equal - length nature of opposite sides.

(b) $\overline{BC}$
(c) The intersection point is the image of $D$ under the 180 - degree rotation about $M$. Opposite sides of a parallelogram are equal and parallel, and opposite angles are equal.