Question

triangle abc is reflected across the x - axis. the result is \\( \triangle a ^ { \prime } b ^ { \prime } c ^ { \prime } \\), 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 reflection. original coordinates \\( \
ightarrow \\) final coordinates \\( a ( - 3, - 2 ) \
ightarrow a ^ { \prime } ( - 3,2 ) \\) \\( b ( 6, - 2 ) \
ightarrow b ^ { \prime } ( 6,2 ) \\) \\( c ( - 1, - 6 ) \
ightarrow c ^ { \prime } ( - 1,6 ) \\) (b) choose the general rule below that describes the reflection mapping \\( \triangle a b c \\) to \\( \triangle a ^ { \prime } b ^ { \prime } c ^ { \prime } \\). \\( ( x, y ) \
ightarrow ( y, x ) \\) \\( ( x, y ) \
ightarrow ( - x, y ) \\) \\( ( x, y ) \
ightarrow ( - y, - x ) \\) \\( ( x, y ) \
ightarrow ( x, - y ) \\)

Explanation:

Step1: Analyze the reflection rule

When a point \((x,y)\) is reflected across the \(x -\)axis, the \(x -\)coordinate remains the same, and the \(y -\)coordinate changes its sign.
For example, if we have a point \(P(x,y)\), its reflection \(P'\) across the \(x -\)axis is \(P'(x,-y)\).

• For point \(A(-3,-2)\): \(x=-3,y = - 2\), after reflection across the \(x -\)axis, \(y\) becomes \(2\), so \(A'(-3,2)\)
• For point \(B(6,-2)\): \(x = 6,y=-2\), after reflection across the \(x -\)axis, \(y\) becomes \(2\), so \(B'(6,2)\)
• For point \(C(-1,-6)\): \(x=-1,y = - 6\), after reflection across the \(x -\)axis, \(y\) becomes \(6\), so \(C'(-1,6)\)

Step2: Check each option for the general rule

• Option \((x,y)\to(y,x)\) represents a reflection across the line \(y = x\).
• Option \((x,y)\to(-x,y)\) represents a reflection across the \(y -\)axis.
• Option \((x,y)\to(-y,-x)\) represents a rotation of \(180^{\circ}\) about the origin (or a reflection across the line \(y=-x\)).
• Option \((x,y)\to(x,-y)\) represents a reflection across the \(x -\)axis.

Answer:

For part (a), the coordinates are filled as \(A(-3,-2)\to A'(-3,2)\), \(B(6,-2)\to B'(6,2)\), \(C(-1,-6)\to C'(-1,6)\) as shown.
For part (b), the general rule is \((x,y)\to(x,-y)\)