Question

triangle \\(xyz\\) is rotated \\(180^{\circ}\\) counterclockwise about the origin. the result is \\(\delta xyz\\), as shown below.

(a) the arrows below show that the coordinates on the left are mapped to the coordinates on the right. fill in the blanks to give the coordinates after the rotation.
original coordinates \\(\
ightarrow\\) final coordinates
\\(x(1, -5) \
ightarrow x(\box, \box)\\)
\\(y(2, 1) \
ightarrow y(\box, \box)\\)
\\(z(4, -7) \
ightarrow z(\box, \box)\\)

(b) choose the general rule below that describes the rotation mapping \\(\delta xyz\\) to \\(\delta xyz\\).
\\(\bigcirc (x, y) \
ightarrow (-x, y)\\)
\\(\bigcirc (x, y) \
ightarrow (y, x)\\)
\\(\bigcirc (x, y) \
ightarrow (-y, -x)\\)
\\(\bigcirc (x, y) \
ightarrow (-y, x)\\)
\\(\bigcirc (x, y) \
ightarrow (y, -x)\\)
\\(\bigcirc (x, y) \
ightarrow (-x, -y)\\)
\\(\bigcirc (x, y) \
ightarrow (x, -y)\\)

Explanation:

⚡ Using what you learned: transformations in the coordinate plane

Step 1: Apply the rotation rule to find the final coordinates

A rotation of \(180^\circ\) counterclockwise about the origin maps any point \((x, y)\) to \((-x, -y)\).

Applying this rule to the given vertices:
\[
X(1, -5)
ightarrow X'(-1, 5)
\]
\[
Y(2, 1)
ightarrow Y'(-2, -1)
\]
\[
Z(4, -7)
ightarrow Z'(-4, 7)
\]

Step 2: Identify the general rule

The transformation rule that represents a \(180^\circ\) rotation about the origin is:
\[
(x, y)
ightarrow (-x, -y)
\]

Answer:

(a) Final coordinates:

• \(X'( -1, 5 )\)
• \(Y'( -2, -1 )\)
• \(Z'( -4, 7 )\)

(b) General rule:

• \((x, y)

ightarrow (-x, -y)\)