Question

triangle ( abc ) is the image of triangle ( abc ) under a translation right 4 units and down 5 units followed by a reflection. write the rules for the translation and reflection. translation: ( (x,y)\to(square,square) ) reflection: ( (x,y)\to(square,square) )

Explanation:

Step1: Determine the translation rule

A translation right \(4\) units and down \(5\) units. For a point \((x,y)\), moving right \(4\) units changes the \(x -\) coordinate: \(x\to x + 4\), and moving down \(5\) units changes the \(y -\) coordinate: \(y\to y-5\). So the translation rule is \((x,y)\to(x + 4,y - 5)\).

Step2: Determine the reflection rule

Let's assume a general point after translation \(P(x',y')=(x + 4,y - 5)\).
We can use a point - by - point analysis. For example, take point \(A(1,3)\). After translation \(A_1(1 + 4,3-5)=(5,-2)\). Let the reflection be over the \(x -\) axis. The rule for reflection over the \(x -\) axis is \((x,y)\to(x,-y)\). If we check another point: \(B(2,1)\), after translation \(B_1(2 + 4,1-5)=(6,-4)\), and after reflection over the \(x -\) axis \(B'(6,4)\) (matches the graph).

Answer:

Translation: \((x,y)\to(x + 4,y - 5)\)
Reflection: \((x,y)\to(x,-y)\)