Question

1. suppose the equation of line $ell$ is $x = 1$. given points $m(3,3)$, $n(4,4)$, and $o(5,2)$, graph $\triangle mno$ and the reflection image $r_{ell}(\triangle mno)$.

Explanation:

Step1: Find the reflection of point \(M(3,3)\)

The formula for reflection over the line \(x = a\) is \((x,y)\to(2a - x,y)\). Here \(a = 1\).
For point \(M(3,3)\), \(x=3,y = 3\). Then \(2a-x=2\times1 - 3=-1\). So \(M'\) is \((-1,3)\).

Step2: Find the reflection of point \(N(4,4)\)

For point \(N(4,4)\), \(x = 4,y=4\). Using the formula \(2a - x=2\times1-4=-2\). So \(N'\) is \((-2,4)\).

Step3: Find the reflection of point \(O(5,2)\)

For point \(O(5,2)\), \(x = 5,y = 2\). Using the formula \(2a - x=2\times1-5=-3\). So \(O'\) is \((-3,2)\).

Step4: Graph \(\triangle MNO\) and \(\triangle M'N'O'\)

Plot the points \(M(3,3)\), \(N(4,4)\), \(O(5,2)\) and connect them to form \(\triangle MNO\).
Plot the points \(M'(-1,3)\), \(N'(-2,4)\), \(O'(-3,2)\) and connect them to form \(r_{\ell}(\triangle MNO)=\triangle M'N'O'\).

Answer:

The reflection of \(M(3,3)\) is \(M'(-1,3)\), the reflection of \(N(4,4)\) is \(N'(-2,4)\) and the reflection of \(O(5,2)\) is \(O'(-3,2)\). Graph \(\triangle MNO\) with vertices \((3,3)\), \((4,4)\), \((5,2)\) and \(\triangle M'N'O'\) with vertices \((-1,3)\), \((-2,4)\), \((-3,2)\) on the coordinate - plane.