Question

1. $\triangle def:d(-4,-5),e(-2,-4),f(-1,0);r_{y = x}circ t_{0,5}$

Explanation:

Step1: Translation

First, perform the translation \(T_{0,5}\). The rule for translation \(T_{0,5}\) is \((x,y)\to(x,y + 5)\).
For point \(D(-4,-5)\): \((-4,-5)\to(-4,-5 + 5)=(-4,0)\)
For point \(E(-2,-4)\): \((-2,-4)\to(-2,-4 + 5)=(-2,1)\)
For point \(F(-1,0)\): \((-1,0)\to(-1,0 + 5)=(-1,5)\)

Step2: Reflection

Next, perform the reflection \(r_{y=x}\). The rule for reflection over the line \(y = x\) is \((x,y)\to(y,x)\)
For the translated point \(D'(-4,0)\): \((-4,0)\to(0,-4)\)
For the translated point \(E'(-2,1)\): \((-2,1)\to(1,-2)\)
For the translated point \(F'(-1,5)\): \((-1,5)\to(5,-1)\)

Answer:

The coordinates of the image of \(\triangle DEF\) after the composition \(r_{y=x}\circ T_{0,5}\) are \(D''(0,-4)\), \(E''(1,-2)\), \(F''(5,-1)\)