Question

example 3:
draw ( overline{mn} ), the image of ( overline{mn} ) after a reflection over the line ( y = -x + 1 ).
( m(2, 4) ) ( n(7, 6) )

iii. describing reflections
example 4: describe the reflections shown below.

example 5: triangle ( abc ) is reflected across the line ( y = 2x ) to form triangle ( rst ).
select all of the true statements.
( overline{ab} = overline{rs} )
( overline{ab} = 2 \bullet overline{rs} )
( \triangle abc sim \triangle rst )
( \triangle abc cong \triangle rst )
( mangle bac = mangle srt )
( mangle bac = 2 \bullet mangle srt )

Explanation:

Example 3

Step1: Find reflection of $M(2,4)$

To reflect a point $(x,y)$ over $y=-x+1$, use the formula for reflection over line $ax+by+c=0$. Rewrite $y=-x+1$ as $x+y-1=0$.
The reflection $(x',y')$ is given by:
$$x' = x - \frac{2a(ax+by+c)}{a^2+b^2}, \quad y' = y - \frac{2b(ax+by+c)}{a^2+b^2}$$
For $M(2,4)$: $ax+by+c=2+4-1=5$, $a=1,b=1$
$$x' = 2 - \frac{2*1*5}{1+1} = 2-5=-3$$
$$y' = 4 - \frac{2*1*5}{1+1} = 4-5=-1$$
So $M'(-3,-1)$

Step2: Find reflection of $N(7,6)$

For $N(7,6)$: $ax+by+c=7+6-1=12$
$$x' = 7 - \frac{2*1*12}{1+1} = 7-12=-5$$
$$y' = 6 - \frac{2*1*12}{1+1} = 6-12=-6$$
So $N'(-5,-6)$

Step3: Plot $M'N'$

Connect points $(-3,-1)$ and $(-5,-6)$ on the grid.

1. Left graph: The top triangle is reflected over the $x$-axis (horizontal axis) to form the bottom triangle, as each point $(x,y)$ maps to $(x,-y)$.
2. Right graph: Triangle $ABC$ is reflected over the origin (point $(0,0)$) to form triangle $PQR$, as each point $(x,y)$ maps to $(-x,-y)$.

A reflection is a rigid transformation, meaning it preserves side lengths, angle measures, and congruence of the figure. Congruent figures are also similar.

• $\overline{AB}=\overline{RS}$: True, reflections preserve side length.
• $\overline{AB}=2\cdot\overline{RS}$: False, no scaling occurs in reflection.
• $\triangle ABC \sim \triangle RST$: True, congruent figures are always similar.
• $\triangle ABC \cong \triangle RST$: True, reflections produce congruent figures.
• $m\angle BAC = m\angle SRT$: True, reflections preserve angle measures.
• $m\angle BAC = 2\cdot m\angle SRT$: False, angles are unchanged.

Answer:

The image $\overline{M'N'}$ has endpoints $M'(-3,-1)$ and $N'(-5,-6)$ (plot and connect these points on the provided coordinate grid).

---

Example 4