Question

the rule ( t_{5,-0.5}circ r_{0,180^{circ}}(x,y) ) is applied to ( \triangle fgh ) to produce ( \triangle fgh ). what are the coordinates of vertex ( f ) of ( \triangle fgh )? ( (4,-1.5) ) ( (4,-0.5) ) ( (-1.5,4) ) ( (-0.5,4) )

Explanation:

Step1: Find the coordinates of \( F \)

From the graph, the coordinates of \( F \) are \( (1,1) \).

Step2: Apply the rotation \( R_{0,180^{\circ}} \)

The rule for a \( 180^{\circ} \) rotation about the origin \( (x,y)\to(-x,-y) \).
For \( (1,1) \), after rotation \( R_{0,180^{\circ}} \), we get \( (-1,-1) \).

Step3: Apply the translation \( T_{5,-0.5} \)

The rule for translation \( T_{a,b}(x,y)=(x + a,y + b) \).
Here \( a = 5,b=-0.5 \), for \( (-1,-1) \), we have \( x=-1+5 = 4,y=-1+( - 0.5)=-1.5 \).

Answer:

\( (4,-1.5) \)