Question

1. define the vertices of $\triangle abc$ to have unique points $a(x_1, y_1)$, $b(x_2, y_2)$, and $c(x_3, y_3)$. given\
2. use rigid transformations to transform $\triangle abc$ to $\triangle abc$ so that $a$ is at the origin and $\overline{ac}$ lies on the x - axis in the positive direction. in the coordinate plane, any point can be moved to any other point using rigid transformations and any line can be moved to any other line using rigid transformations.\
3. any property that is true for $\triangle abc$ will also be true for $\triangle abc$. definition of congruence\
4. let $r$, $s$, and $t$ be real numbers such that the vertices of $\triangle abc$ are $a(0, 0)$, $b(2r, 2s)$, and $c(2t, 0)$. defining constants\
5. let $d$, $e$, and $f$ be the midpoints of $\overline{ab}$, $\overline{bc}$ and $\overline{ac}$ respectively. defining points\
6. $d = (r, s)$\

$e = (r + t, s)$\
$f = (t, 0)$ definition of midpoints\

1. slope of $\overline{ae} = \frac{s}{r + t}$\

slope of $\overline{bf} = \frac{2s}{2r - t}$\
slope of $\overline{cd} = \frac{-s}{2t - r}$\
21\
what is the reason for statement 7 in the given proof?\
a. definition of midpoint\
b. definition of slope\
c. parallel lines have equal slopes.\
d. using point - slope formula

Explanation:

Statement 7 calculates the slopes of three line segments using their endpoint coordinates. The formula for the slope between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $\frac{y_2-y_1}{x_2-x_1}$, which is the core definition of slope. This matches the calculations shown.

Answer:

B. definition of slope