Question

triangle abc is reflected across the x - axis.
the result is △abc, 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 → final coordinates
a(-3,2)→a(□,□)
b(7,10)→b(□,□)
c(7,4)→c(□,□)
(b) choose the general rule below that describes
the reflection mapping △abc to △abc.
(x,y)→(-y,-x) (x,y)→(-x,y)
(x,y)→(y,-x) (x,y)→(-y,x)
(x,y)→(y,x) (x,y)→(x,-y)
(x,y)→(-x,-y)

Explanation:

Step1: Recall the rule for reflection across the x - axis

When a point \((x,y)\) is reflected across the \(x\) - axis, the \(x\) - coordinate remains the same, and the \(y\) - coordinate changes its sign. The rule is \((x,y)\to(x, - y)\).

Step2: Apply the rule to point \(A(-3,2)\)

For \(A(-3,2)\), using the rule \((x,y)\to(x, - y)\), we substitute \(x=-3\) and \(y = 2\). So \(A'(-3,-2)\).

Step3: Apply the rule to point \(B(7,10)\)

For \(B(7,10)\), substitute \(x = 7\) and \(y=10\) into the rule \((x,y)\to(x, - y)\). So \(B'(7,-10)\).

Step4: Apply the rule to point \(C(7,4)\)

For \(C(7,4)\), substitute \(x = 7\) and \(y = 4\) into the rule \((x,y)\to(x, - y)\). So \(C'(7,-4)\).

Step5: Verify the general rule

We have shown that for any point \((x,y)\) in \(\triangle ABC\), its image \((x',y')\) in \(\triangle A'B'C'\) satisfies \(x'=x\) and \(y'=-y\). So the rule is \((x,y)\to(x, - y)\)

Answer:

a) \(A(-3,2)\to A'(-3,-2)\), \(B(7,10)\to B'(7,-10)\), \(C(7,4)\to C'(7,-4)\)
b) \((x,y)\to(x, - y)\)