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(\square, \square))
(y(2, 1) \
ightarrow y(\square, \square))
(z(4, -7) \
ightarrow z(\square, \square))

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

Explanation:

🆕 New Concept Discovered: Coordinate Rotations
Rotating points changes their signs or swaps coordinates.

Step 1: Apply the rotation rule for 180 degrees

A rotation of \(180^\circ\) (either clockwise or counterclockwise) about the origin maps any point \((x, y)\) to \((-x, -y)\). This means we change the sign of both the \(x\)-coordinate and the \(y\)-coordinate.

\[
(x, y)
ightarrow (-x, -y)
\]

Step 2: Find the final coordinates for each vertex

Using the rule \((x, y)
ightarrow (-x, -y)\), we transform the original coordinates:

• For \(X(1, -5)\):

\[
X'( -1, -(-5) )
ightarrow X'(-1, 5)
\]

• For \(Y(2, 1)\):

\[
Y'( -2, -1 )
ightarrow Y'(-2, -1)
\]

• For \(Z(4, -7)\):

\[
Z'( -4, -(-7) )
ightarrow Z'(-4, 7)
\]

Step 3: Identify the general rule

From the transformation applied above, the general rule that describes a \(180^\circ\) rotation about the origin is:

\[
(x, y)
ightarrow (-x, -y)
\]

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Answer:

(a) Final Coordinates:

• \(X'(1, -5)

ightarrow X'(-1, 5)\)

• \(Y'(2, 1)

ightarrow Y'(-2, -1)\)

• \(Z'(4, -7)

ightarrow Z'(-4, 7)\)

(b) General Rule:

• \((x, y)

ightarrow (-x, -y)\)