I have a homework assignment that I just can't solve.
The point (3.4) is reflected on the line $y = 2x +1$. Which coordinates are the mirror image.
I know that I have to use the following formula $y = kx+m$ but do not know how.
Thanks!!
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Do you know the condition for two lines to be perpendicular? What are $k_1,k_2$ supposed to be? – almagest Sep 11 '14 at 14:25
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- Find the equation of the line through (3, 4) which is perpendicular to the given line. 2. Find where the two lines intersect (X, Y). 3. If (a, b) is your mirrored point then (a+3)/2 = X and (b+4)/2 = Y.
– Paul Sep 11 '14 at 14:43
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Actually, you have to draw a perpendicular line to the original line which includes $P(3,4)$. The condition for perpendicular lines is $k_1\cdot k_2=-1$. Therefore, perpendicular line should be:
$$y=\frac{-1}{2}x+m$$
You can find $m=\frac{11}{2}$ by putting $P(3,4)$.
Now you have to find the intersection point $P(x_c,y_c)$ by calculating:
$$2x_c+1=\frac{-1}{2}x_c+\frac{11}{2}$$
Then $P(x_c,y_c)$ will be the center of your mirror. You can find the reflection with the middle point formulas:
$$x_c=\frac{x_2+x_1}{2} \;\;\;\; y_c=\frac{y_2+y_1}{2}$$
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There are 3 paralel lines here. Sketch or plot line parallel to given line through given point. i.e.,solve for unknown c in y = 2 x + c. Observe y-intercepts and see how the intercept of given line can be made exactly in between c of this line and c of the required line.
By this procedure, y = 2 x - 2 contains given point (3,4), y = 2 x + 4 is the required line mirrored by given line y = 2 x + 1. Their y- intercepts (-2,1,4) are in arithmetic progression.
Narasimham
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