Write the equation of the mirror line for the segment with endpoints (5, 4) and (-1, -1).

Find the reflection of the point A(3, -2) through the mirror line of y = 1/2x − 1.

Question

Write the equation of the mirror line for the segment with endpoints (5, 4) and (-1, -1).

Find the reflection of the point A(3, -2) through the mirror line of y = 1/2x − 1.

polaris_s0i · Answer

first find you're slope:

$$ m = \frac{\Delta y}{\Delta x}$$
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
$$ m = \frac{4 + 1}{5 + 1} = \frac{5}{6}$$

then substitute a point into the equation:
$$y = \frac{5}{6}x + b$$
$$4 = \frac{5}{6}5 + b$$
$$4 = \frac{25}{6} + b$$
$$4 - \frac{25}{6} = b$$
$$ b = -\frac{1}{6}$$

so the equation of the mirror line is:
$$y=\frac{5}{6}x - \frac{1}{6}$$

anonymous · Answer

wow thank you so much!o: 
do you know how to do the second one? Or can i do it in the same way you did the first one?

polaris_s0i · Answer

second one is different, one sec.

polaris_s0i · Answer

for reflection of (3,-2), you need to find the distance to the line:
$$ y = \frac{1}{2}x - 1$$

the "shortest" distance is on a line perpendicular to this one.
perpendicular lines have a slope that is the negative reciprocal of the slope of the original

so:

$$ y = -2x + b $$

plug in (3, -2) to find b:

$$ -2 = -2(3) + b $$
$$ -2 = -6 + b $$
$$ 4 = b $$

$$ y = -2x + 4 $$
now you find the point where they intersect:

$$ -2x + 4 = \frac{1}{2}x - 1 $$
$$ -2x - \frac{1}{2}x = -1 - 4$$
$$\frac{-4}{2}x - \frac{1}{2}x = -5$$
$$\frac{-5}{2}x = -5$$
$$x = 2$$

the y coordinate is found by substituting x back into one of the lines (doesn't matter which one).

$$y = -2*2 + 4$$
$$y = -4 + 4$$
$$y = 0 $$

so the midpoint is (2, 0)

this means we went one unit left in x to get to the midpoint, so we can go one more unit left to get to the reflection.  So lets plug in 1 for x.

$$y = -2(1) + 4$$
$$y = 2$$

so the reflection point should be (1, 2)

polaris_s0i · Answer

to check take calculate the distance between the midpoint and A:

$$s^2 = (\Delta x)^2 + (\Delta y)^2 $$
$$s^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2$$
$$s^2 = (3 - 2)^2 + (-2 - 0)^2$$
$$s^2 = 1 + 4$$
$$s = \sqrt{5}$$

then calculate the distance from the midpoint to the reflected point:

$$s^2 = (\Delta x)^2 + (\Delta y)^2 $$
$$s^2 = (x_3 - x_2)^2 + (y_3 - y_2)^2$$
$$s^2 = (2 - 1)^2 + (0 - 2)^2$$
$$s^2 = 1 + 4$$
$$s = \sqrt{5}$$

they are the same, so we know we are correct!

anonymous · Answer

Thank you so much for these you're awesome! :3