The point A (7, 6) is reflected about the line x = 3, then about the line x = k. The final image is A”(5, 6). What is the value of k?
reflection about x=3 takes (x,y) -> (3-x,y)
reflection about x=k takes (x,y) -> (k-x,y)
So, both reflections take
(x,y)->(3-x,y)->(k-(3-x),y) = (k-3+x,y)
So, since (7,6)->(5,6)
k-3+7 = 5
k=1
To determine the value of k, we need to understand the reflection process.
First, let's find the reflection of point A (7, 6) about the line x = 3.
Step 1: Determine the distance between the point and the line.
- The distance between a point (x, y) and the line x = a is given by |x - a|.
So, the distance between point A and the line x = 3 is |7 - 3| = 4.
Step 2: Reflect the point across the line.
- When we reflect a point across a vertical line, we need to change only the x-coordinate. The y-coordinate remains the same.
Since the line x = 3 is vertical, the x-coordinate of point A' (the reflection of A about the line x = 3) will be: 3 - 4 = -1.
Therefore, the coordinates of A' are (-1, 6).
Next, let's find the reflection of point A' (-1, 6) about the line x = k.
We are given that the final image is A" (5, 6).
Step 1: Determine the distance between the point and the line.
- The distance between a point (x, y) and the line x = a is given by |x - a|.
So, the distance between point A' and the line x = k is |-1 - k|.
Step 2: Reflect the point across the line.
- When we reflect a point across a vertical line, we need to change only the x-coordinate. The y-coordinate remains the same.
So, the x-coordinate of A" should be: k - |-1 - k| = k + |k + 1|.
Since the final image A" is (5, 6), this means that k + |k + 1| = 5.
Therefore, to find the value of k, we can solve the equation k + |k + 1| = 5.
To find the value of k in this problem, we can use the concept of reflections in coordinate geometry.
First, let's consider the reflection of point A (7, 6) about the line x = 3. When a point is reflected about a vertical line, its x-coordinate remains the same, but its y-coordinate changes to its opposite value. In this case, since the line x = 3 is a vertical line, the reflection of point A becomes A' (3, -6).
Next, we need to reflect A' (3, -6) about the line x = k. We know that the final image of A' after this reflection is A" (5, 6).
To determine the value of k, we need to figure out how the reflection about the line x = k transforms the x-coordinates of the points.
When reflecting a point about a vertical line x = k, the x-coordinate of the point remains the same, while the y-coordinate remains the opposite. Therefore, A" (5, 6) is the reflection of A' (3, -6) about the line x = k.
Since the x-coordinate of A" is 5, we can infer that the line of reflection x = k passes through the point A" (5, 6). Therefore, the value of k is 5.
Hence, the value of k in this problem is 5.