The triangle with vertices at A (-2, 2), B (-8, -1) and C (-8, -1) is reflected about the line y= 2x + 1. Express the coordinates of the reflection of A as an ordered pair.
How do you set this problem up?
Thank you.
A neat way to check your thinking. Draw the system (triangle, line) on a piece of graph paper. Fold the paper along the line. That is reflection.
Now. The math:
http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node9.html
First, if B and C have the same coordinates then we have a line segment not a triangle.
To find the reflection of a point in a line, you can follow these steps:
1. Find the equation of the line of reflection: In this case, the equation of the line is given as y = 2x + 1.
2. Find the perpendicular distance from the point A to the line of reflection: You can use the formula for the distance between a point and a line to find this distance. The formula is:
d = |Ax + By + C| / sqrt(A^2 + B^2)
In our case, the equation of the line is y = 2x + 1, which can be written as 2x - y + 1 = 0. Comparing this with the general form Ax + By + C = 0, we get A = 2, B = -1, and C = 1. Plugging these values into the formula, we get:
d = |(2)(-2) + (-1)(2) + 1| / sqrt((2)^2 + (-1)^2)
= |-4 - 2 + 1| / sqrt(4 + 1)
= |-5| / sqrt(5)
= 5 / sqrt(5)
= sqrt(5)
3. Find the midpoint of the line segment connecting the point A and its projection on the line of reflection: The projection is the point on the line of reflection that is closest to point A. To find this point, you can find the intersection point between the line of reflection and the line perpendicular to it passing through point A.
The perpendicular line to y = 2x + 1 can be found by taking the negative reciprocal of its slope, which gives -1/2. Since the point A (-2, 2) lies on this line, we can use the point-slope form of a line to find its equation:
y - 2 = (-1/2)(x + 2)
y - 2 = (-1/2)x - 1
y = (-1/2)x + 1
Now, we want to find the intersection of this line with y = 2x + 1. So we can set them equal to each other:
(-1/2)x + 1 = 2x + 1
(-1/2)x - 2x = 1 - 1
-5/2 x = 0
x = 0
Plugging this value back into y = 2x + 1, we get:
y = 2(0) + 1
y = 1
Therefore, the projection of point A on the line of reflection is (0, 1).
Now, we can find the midpoint of the line segment connecting A and (0, 1) by taking the average of their x and y coordinates:
Midpoint (m_x, m_y):
m_x = (-2 + 0) / 2 = -1
m_y = (2 + 1) / 2 = 3/2 = 1.5
4. Find the reflection point: The reflection point is obtained by finding the point symmetric to point A with respect to the midpoint.
We can find the reflection point by using the formula:
Reflection point (R_x, R_y):
R_x = 2 * m_x - A_x
= 2 * -1 - (-2)
= -2 + 2
= 0
R_y = 2 * m_y - A_y
= 2 * 1.5 - 2
= 3 - 2
= 1
Therefore, the reflection of point A (-2, 2) about the line y = 2x + 1 is (0, 1).