Question

suppose the line of reflection is y = 2. what are the coordinates of c’ of δa’b’c’? image of coordinate grid with triangle abc, then options: (-3,5), (-1,-4), (2,5), (-1,3)

Explanation:

Step1: Find coordinates of C

From the graph, point C has coordinates \((-3, -1)\) (assuming the grid and position, let's confirm: looking at the grid, C is at x=-3, y=-1? Wait, no, wait the original triangle: let's check the y-coordinate. Wait, the line of reflection is y=2. Let's first find the original coordinates of C. Looking at the graph, C is at (-3, -1)? Wait, no, maybe I misread. Wait, the y-axis: the original C, let's see the grid. Let's assume the original C is at (-3, -1)? Wait, no, maybe the original C is at (-3, -1)? Wait, no, let's check the distance from y=2.

Wait, reflection over horizontal line y = k: the formula is \((x, y) \to (x, 2k - y)\). So first, find the original coordinates of C. From the graph, C is at (-3, -1)? Wait, no, looking at the grid, the point C: let's see, the x-coordinate is -3, y-coordinate: the line where C is, is y=-1? Wait, no, the original triangle: A is at (-1,1), B at (1,-1), C at (-3,-1)? Wait, no, maybe I made a mistake. Wait, the line of reflection is y=2. Let's find the original y-coordinate of C. Let's say original C is (x, y) = (-3, -1). Then the reflection over y=2: the formula is (x, 22 - y) = (x, 4 - y). So y-coordinate of C is -1, so 4 - (-1) = 5? Wait, no, 22 is 4, 4 - (-1) = 5? Wait, no, reflection over y=k: the distance from the point to the line is |y - k|, so the reflected point is at y = k + (k - y) = 2k - y. So if original y is -1, k=2, then reflected y is 2*2 - (-1) = 4 +1 =5. And x-coordinate remains the same, -3. So C' is (-3,5). Let's check the options: (-3,5) is one of the options.

Step2: Apply reflection formula

Reflection over horizontal line \( y = k \): for a point \( (x, y) \), the reflected point \( (x, y') \) is given by \( y' = 2k - y \). Here, \( k = 2 \), original coordinates of C: from the graph, C is at \( (-3, -1) \) (x=-3, y=-1). So \( y' = 2*2 - (-1) = 4 + 1 = 5 \). So the coordinates of C' are \( (-3, 5) \).

Answer:

\((-3, 5)\) (corresponding to the option: (-3,5))