1. **Problem statement:** Find the vertices of the triangle invariant under two transformations: a) Reflection in the line $y=x$. b) Reflection in the line $y=x$ followed by a 180° rotation about the point $(5,5)$. 2. **Recall the vertices:** $A=(3,7), B=(7,3), C=(3,3)$. 3. **Part (a): Reflection in the line $y=x$** - Reflection rule: A point $(x,y)$ reflected about $y=x$ becomes $(y,x)$. - A vertex is invariant if it maps to itself, so $(x,y) = (y,x)$ which implies $x=y$. Check each vertex: - $A=(3,7) \to (7,3)$, not invariant. - $B=(7,3) \to (3,7)$, not invariant. - $C=(3,3) \to (3,3)$, invariant. So, only vertex $C$ is invariant under reflection in $y=x$. 4. **Part (b): Reflection in $y=x$ followed by 180° rotation about $(5,5)$** - Step 1: Reflect $(x,y)$ about $y=x$ to get $(y,x)$. - Step 2: Rotate 180° about $(5,5)$. Rotation formula about $(h,k)$ by 180°: $$ (x',y') = (2h - x, 2k - y) $$ Apply to $(y,x)$: $$ (x'', y'') = (2\cdot5 - y, 2\cdot5 - x) = (10 - y, 10 - x) $$ A vertex is invariant if: $$ (x,y) = (10 - y, 10 - x) $$ This gives the system: $$ x = 10 - y $$ $$ y = 10 - x $$ From the first equation, $y = 10 - x$. Substitute into the second: $$ y = 10 - x = 10 - x $$ This is consistent. So the invariant points satisfy: $$ y = 10 - x $$ Check each vertex: - $A=(3,7)$: $7 \stackrel{?}{=} 10 - 3 = 7$, yes invariant. - $B=(7,3)$: $3 \stackrel{?}{=} 10 - 7 = 3$, yes invariant. - $C=(3,3)$: $3 \stackrel{?}{=} 10 - 3 = 7$, no. So vertices $A$ and $B$ are invariant under the combined transformation. **Final answers:** - (a) Invariant vertex: $C(3,3)$ - (b) Invariant vertices: $A(3,7)$ and $B(7,3)$