Question

1. line segment rs has endpoints r (-2, 4) and s (-4, -1). line segment rs has endpoints r (3, -3) and s (5, 2). name the rigid transformations that map line segment rs to line segment rs. sequence: r (-2, 4) → r (□, □) → r (3, -3) s (-4, -1) → s (□, □) → s (5, 2)

Explanation:

Step1: Analyze the transformation from \(R(-2,4)\) to \(R''(3,-3)\) and \(S(-4,-1)\) to \(S''(5,2)\)

First, consider the \(x\) - coordinates:
For \(R\): \(-2\) to \(3\), the change is \(3-(-2)=5\).
For \(S\): \(-4\) to \(5\), the change is \(5 - (-4)=9\). This is not a simple translation.
Now, consider a reflection. If we reflect \(R(-2,4)\) and \(S(-4,-1)\) over the \(y\) - axis, the new points \(R'(2,4)\) and \(S'(4,-1)\) (using the rule \((x,y)\to(-x,y)\)).
Then, for the \(x\) - coordinate of \(R'\) (after reflection \(x = 2\)) to \(R''(3)\): \(3 - 2=1\). For the \(y\) - coordinate of \(R'\) (after reflection \(y = 4\)) to \(R''(-3)\): \(-3-4=-7\).
For the \(x\) - coordinate of \(S'\) (after reflection \(x = 4\)) to \(S''(5)\): \(5 - 4 = 1\). For the \(y\) - coordinate of \(S'\) (after reflection \(y=-1\)) to \(S''(2)\): \(2-(-1)=3\). This is not correct.
If we reflect over the \(x\) - axis (using the rule \((x,y)\to(x,-y)\)), \(R(-2,4)\to R'(-2,-4)\) and \(S(-4,-1)\to S'(-4,1)\)
Then, for the \(x\) - coordinate of \(R'\) to \(R''(3)\): \(3-(-2)=5\). For the \(y\) - coordinate of \(R'\) to \(R''(-3)\): \(-3-(-4) = 1\).
For the \(x\) - coordinate of \(S'\) to \(S''(5)\): \(5-(-4)=9\). This is wrong.
Let's try a rotation. Rotate \(180^{\circ}\) about the origin (using the rule \((x,y)\to(-x,-y)\)), \(R(-2,4)\to R'(2,-4)\) and \(S(-4,-1)\to S'(4,1)\)
Then, for the \(x\) - coordinate: \(3 - 2=1\), for the \(y\) - coordinate: \(-3-(-4)=1\) for \(R\). For \(S\), \(5 - 4 = 1\) and \(2 - 1=1\)
The translation rule is \((x,y)\to(x + 1,y+1)\)

Step2: Verify the transformation

After rotation \(180^{\circ}\) about the origin:
\(R(-2,4)\to R'(2,-4)\), then apply translation \((x,y)\to(x + 1,y + 1)\), \(R'(2,-4)\to R''(2 + 1,-4+1)=(3,-3)\)
\(S(-4,-1)\to S'(4,1)\), then apply translation \((x,y)\to(x + 1,y + 1)\), \(S'(4,1)\to S''(4 + 1,1+1)=(5,2)\)

Answer:

Sequence: Rotation \(180^{\circ}\) about the origin followed by translation \((x,y)\to(x + 1,y + 1)\)
\(R(-2,4)\to R'(2,-4)\to R''(3,-3)\)
\(S(-4,-1)\to S'(4,1)\to S''(5,2)\)