Question

1. in the grid below, points a and b have been rotated 90° counter - clockwise around the origin. give their image coordinates.

a(1,3)→( ) b(5,6)→( )

1. which of the following algebraic rules is equivalent to the rotation in #4?

(1) (x,y)→(y,x)
(2) (x,y)→( - x,y)
(3) (x,y)→( - y,x)
(4) (x,y)→( - x,y)

1. find the image of c using the same rotation. plot it and finish drawing △abc.

reasoning

1. 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, (overline{ac}), is drawn and its midpoint m is located. use tracing paper in this problem to help you answer the following questions.

(a) if (overline{cd}) was rotated 180° about m, explain why it would have to lie on top of (overline{ab}).
(b) for the same reason, if (overline{ad}) was rotated 180° about m, what line would it lie on top of?
(c) at what point would the images of (overline{cd}) and (overline{ad}) have to intersect? based on this, what can you conclude about opposite sides and opposite angles of a parallelogram? explain your answer.

Explanation:

Step1: Recall 90 - degree counter - clockwise rotation rule

The rule for a 90 - degree counter - clockwise rotation about the origin is $(x,y)\to(-y,x)$.

Step2: Apply the rule to point A

For point $A(1,3)$, substituting $x = 1$ and $y = 3$ into the rule $(x,y)\to(-y,x)$, we get $(-3,1)$.

Step3: Apply the rule to point B

For point $B(5,6)$, substituting $x = 5$ and $y = 6$ into the rule $(x,y)\to(-y,x)$, we get $(-6,5)$.

Step4: Identify the correct rotation rule for question 5

The rule $(x,y)\to(-y,x)$ is the 90 - degree counter - clockwise rotation rule. So the answer to question 5 is (3).

Step5: Answer 7(a)

In a parallelogram $ABCD$ with diagonal $AC$ and mid - point $M$ of $AC$, since $AB\parallel CD$ and $M$ is the mid - point of $AC$, a 180 - degree rotation about $M$ maps $CD$ onto $AB$ because the mid - point of the diagonal of a parallelogram is the center of rotational symmetry for the parallelogram. The vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$ are equal in magnitude and opposite in direction, and a 180 - degree rotation about the mid - point of the diagonal that connects non - parallel vertices will map one side to the other.

Step6: Answer 7(b)

If $\overline{AD}$ is rotated 180 degrees about $M$, it will lie on top of $\overline{BC}$ because in a parallelogram $AD\parallel BC$ and the center of rotational symmetry (mid - point of the diagonal $AC$) will map one parallel side to the other under a 180 - degree rotation.

Step7: Answer 7(c)

The images of $\overline{CD}$ and $\overline{AD}$ will intersect at point $A$. We can conclude that opposite sides of a parallelogram are equal in length and parallel, and opposite angles are equal. This is because of the rotational symmetry of the parallelogram about the mid - point of its diagonals. A 180 - degree rotation about the mid - point of a diagonal maps one side to its opposite side, showing congruence of opposite sides. Also, since parallel lines are mapped to parallel lines under rotation, the angle relationships are preserved, showing that opposite angles are equal.

Answer:

1. $A(1,3)\to(-3,1)$; $B(5,6)\to(-6,5)$
2. (3) $(x,y)\to(-y,x)$
3. (a) Because in a parallelogram, the mid - point of the diagonal is the center of rotational symmetry and $\overrightarrow{AB}$ and $\overrightarrow{CD}$ are equal in magnitude and opposite in direction.

(b) $\overline{BC}$
(c) Point $A$. Opposite sides are equal in length and parallel, and opposite angles are equal due to rotational symmetry about the mid - point of the diagonals.