Question

11 pentagon abcde is rotated 180⁰ clockwise about the origin to form pentagon abcde. image of coordinate grid with pentagon which statement is true?
a pentagon abcde is congruent to pentagon abcde.
b the sum of the angle measures of pentagon abcde is 180⁰ more than the sum of the angle measures of pentagon abcde.
c each side length of pentagon abcde is 2 times the corresponding side length of pentagon abcde.
d each side length of pentagon abcde is \\(\frac{1}{2}\\) the corresponding side length of pentagon abcde.
12 a figure is graphed on a coordinate grid as shown.
the figure is rotated 180 degrees with the origin as the center of rotation.
which rule describes this transformation?
a \\((x, y) \
ightarrow (-x, -y)\\)
b \\((x, y) \
ightarrow (x, -y)\\)
c \\((x, y) \
ightarrow (-y, -x)\\)
d \\((x, y) \
ightarrow (-x, y)\\)

Explanation:

Question 11

• Option A: A rotation is a rigid transformation, which means the shape and size of the figure remain unchanged. So, pentagon \( ABCDE \) and its image \( A'B'C'D'E' \) after a \( 180^\circ \) rotation are congruent.
• Option B: The sum of the interior angles of a pentagon is determined by the formula \( (n - 2)\times180^\circ \), where \( n = 5 \). Rotation does not change the number of sides or the angle measures, so the sum of the angles remains the same. Thus, this option is false.
• Option C and D: Rotation is a rigid transformation, so side lengths do not change. Scaling (resizing) changes side lengths, but rotation does not. So these options are false.

The rule for a \( 180^\circ \) rotation about the origin is that the coordinates \( (x, y) \) of a point are transformed to \( (-x, -y) \).

• Option B is the rule for a reflection over the \( x \)-axis.
• Option C is not the standard rule for a \( 180^\circ \) rotation.
• Option D is the rule for a reflection over the \( y \)-axis.

Answer:

A. Pentagon \( ABCDE \) is congruent to pentagon \( A'B'C'D'E' \).

Question 12