Question

1. segment fg: f(1,1), g(6,7); reflection: x - axis; rotation: 180° about origin
2. segment rs: r(2,-1), s(6,-5); translation: <3,4>; reflection: x = 2

Explanation:

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Problem 7:

Step1: Reflect over x-axis

For a point $(x,y)$, reflection over x-axis gives $(x,-y)$.
$F(1,1) \to F'(1,-1)$; $G(6,7) \to G'(6,-7)$

Step2: Rotate 180° about origin

For a point $(x,y)$, 180° rotation gives $(-x,-y)$.
$F'(1,-1) \to F''(-1,1)$; $G'(6,-7) \to G''(-6,7)$

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Problem 8:

Step1: Apply translation $<3,4>$

For a point $(x,y)$, translation $$ gives $(x+a,y+b)$.
$R(2,-1) \to R'(2+3, -1+4)=(5,3)$; $S(6,-5) \to S'(6+3, -5+4)=(9,-1)$

Step2: Reflect over $x=2$

For a point $(x,y)$, reflection over $x=h$ gives $(2h-x,y)$. Here $h=2$.
$R'(5,3) \to R''(2*2-5, 3)=(-1,3)$; $S'(9,-1) \to S''(2*2-9, -1)=(-5,-1)$

Answer:

1. Final endpoints of segment FG after transformations: $F''(-1,1)$, $G''(-6,7)$
2. Final endpoints of segment RS after transformations: $R''(-1,3)$, $S''(-5,-1)$