The point (2.2) is reflected in the line y = -0.5x + 2. Determine the coordinates of the mirror image.
what does mirror image mean? the opposite on the other side? how would that work in this case
thankyou
Make sketch of the line y = (-1/2)x + 2, and plot the point A(2,2)
notice that (2,2) does not lie on the line, it is above it
The image of the reflection A would be a point B(a,b) so that AB
has the given line as its right-bisector.
How about we find the equation of AB, we know its slope is +2 and it passes
through (2,2)
y = 2x + b, at (2,2) ---> 2 = 4 + b
b = -2
equation of AB is y = 2x -2
it meets the given line when 2x - 2 = (-1/2)x + 2
4x - 4 = -x + 4
5x = 8
x = 8/5 , then y = 16/5 - 2 = 6/5
Clearly, (8/5 , 6/5) must be the midpoint of AB, that is
(a+2)/2 = 8/5 and (b+2)/2 = 6/5
a+2 = 16/5 and b+2 = 12/5
a = 6/5 and b = 2/5
so the image of (2,2) after a reflection in y = (-1/2)x + 2 is ........
the opposite on the other side
you want to find a point the same distance on the other side of the line. It must also lie on a line perpendicular to your given line.
google can provide you with many discussions and examples of reflection through any line.
Yes, a mirror image refers to the reflection of a point across a line, resulting in a point that is equidistant from the line but on the opposite side. In this case, the line y = -0.5x + 2 serves as the mirror or reflection line. To find the coordinates of the mirror image, we need to determine the line of reflection and use the formula for reflecting a point across a line.
1. Given point: (2.2)
2. Line of reflection: y = -0.5x + 2
To reflect a point across a line, follow these steps:
Step 1: Find the slope of the line given by the equation y = -0.5x + 2.
The equation is in the form y = mx + b, where m is the slope. In this case, the slope is -0.5.
Step 2: Find the perpendicular slope.
Perpendicular slopes have negative reciprocals. So, the perpendicular slope would be the negative reciprocal of -0.5, which is 2.
Step 3: Find the equation of the line that passes through the given point (2.2) with the perpendicular slope.
Using the point-slope form of a line, the equation of the new line passing through (2.2) with a slope of 2 can be written as:
y - 2.2 = 2(x - 2)
Step 4: Simplify the equation from step 3.
y - 2.2 = 2x - 4
y = 2x - 1.8
Step 5: Find the intersection point of the original line and the new line.
Set the two equations equal to each other and solve for x:
-0.5x + 2 = 2x - 1.8
-0.5x - 2x = -1.8 - 2
-2.5x = -3.8
x = -3.8 / -2.5
x = 1.52
Step 6: Substitute the x-value found in step 5 into either equation to find the y-coordinate of the intersection point.
Using y = -0.5x + 2:
y = -0.5(1.52) + 2
y = -0.76 + 2
y = 1.24
Therefore, the coordinates of the mirror image are (1.52, 1.24).
Yes, in this case, the mirror image refers to the reflection of a point across the given line. It means that the resulting point will be on the opposite side of the line, maintaining the same distance from the line as the original point.
To determine the coordinates of the mirror image, we can use the concept of the perpendicular distance between a point and a line.
Here's how you can find the mirror image of the point (2.2) across the line y = -0.5x + 2:
1. Find the perpendicular distance between the point (2.2) and the line y = -0.5x + 2. To do this, we'll use the formula for the perpendicular distance from a point to a line:
Distance = |Ax + By + C| / sqrt(A^2 + B^2),
where the line equation is in the form Ax + By + C = 0, and (x, y) is the point.
In this case, the coefficients are A = 0.5, B = 1, and C = -2. Plugging in the values, we have:
Distance = |0.5(2) + 1(2.2) - 2| / sqrt(0.5^2 + 1^2)
= |1 + 2.2 - 2| / sqrt(0.25 + 1)
= |1.2| / sqrt(1.25)
= 1.2 / 1.118
= 1.073.
2. Since the point (2.2) is above the line y = -0.5x + 2, the mirror image point will be located below the line, maintaining the same perpendicular distance of 1.073 units.
3. To determine the coordinates of the mirror image point, we subtract the distance from the y-coordinate of the original point (2.2).
Mirror image y-coordinate = 2.2 - 1.073
= 1.127.
Since the mirror image point is reflected across the line, its x-coordinate will remain the same.
Mirror image x-coordinate = 2.2.
Therefore, the coordinates of the mirror image point are approximately (2.2, 1.127).