Find the coordinates of the image of the point (4,2) when it is reflected across the line y=3
y=3 is a horizontal line, the point (4,2) is one unit below it, so reflecting it would put it one unit above it, so
(4,2) ----> (4,4)
Reflect point -4,5 along y =4 followed by y =1
To find the coordinates of the image of the point (4,2) when it is reflected across the line y=3, follow these steps:
Step 1: Find the distance between the point (4,2) and the line y=3. This distance is the perpendicular distance between the point and the line.
To find the perpendicular distance between a point and a line, you can use the formula:
distance = |Ax + By + C| / (sqrt(A^2 + B^2))
In this case, the line y=3 can be written as x + 0y + -3 = 0.
So, A = 1, B = 0, and C = -3.
Substituting the values, we get:
distance = |1 * 4 + 0 * 2 + -3| / (sqrt(1^2 + 0^2))
= |4 - 3| / (sqrt(1))
= 1 / 1
= 1
Therefore, the distance between the point (4,2) and the line y=3 is 1.
Step 2: Use the distance calculated in Step 1 to find the image point.
Since the line y=3 is a horizontal line, the image point will have the same x-coordinate as the original point, but the y-coordinate will be reflected across the line y=3.
The y-coordinate of the image point will be equal to the distance between the line y=3 and the original point, added to 3 (the equation of the line).
So, the y-coordinate of the image point = 3 + 1 (distance calculated in Step 1)
= 4
Therefore, the image point of (4,2) when reflected across the line y=3 is (4,4).
To find the coordinates of the image of the point (4,2) when it is reflected across the line y=3, you can follow these steps:
1. Draw a horizontal line representing the line y=3 on a coordinate plane.
2. Plot the point (4,2) on the same plane.
3. Draw a vertical line connecting the point (4,2) to the line y=3. This line will be perpendicular to the line y=3.
4. Extend the line y=3 to intersect with the perpendicular line you just drew.
5. The point of intersection is the image of the point (4,2) when reflected across the line y=3.
To find the coordinates of the image, you can determine the x-coordinate and the y-coordinate separately.
For the x-coordinate:
- Count the number of units from the line y=3 to the point (4,2).
- Since the point (4,2) is below the line y=3, you will have to go the same magnitude of units in the opposite direction.
- Start at the point of intersection of the line y=3 and the perpendicular line you drew. Move the same number of units in the opposite direction.
- The x-coordinate of the reflected point will be the x-coordinate of the point of intersection minus the number of units you counted.
For the y-coordinate:
- Since the line y=3 is horizontal, the y-coordinate of the reflected point will remain the same.
By following these steps, you can determine the coordinates of the image of the point (4,2) when it is reflected across the line y=3.