The triangle A(1,2),B(4,4) and C(3,7) is first reflected in the line y = 0 onto triangle A'B'C' and then triangle A'B'C' is reflected in the origin onto triangle A''B''C'' . Write down the co-ordinates of :
A'' , B'' and C''

Answer

To solve the problem, we will follow the steps of reflection as described in the question.

### Step 1: Reflect the triangle in the line y = 0

The coordinates of the triangle are:
- A(1, 2)
- B(4, 4)
- C(3, 7)

When reflecting a point (x, y) in the line y = 0, the x-coordinate remains the same, while the y-coordinate changes sign. Thus, the reflected points A', B', and C' will be:

- A' = (1, -2) (y-coordinate changes from 2 to -2)
- B' = (4, -4) (y-coordinate changes from 4 to -4)
- C' = (3, -7) (y-coordinate changes from 7 to -7)

### Step 2: Reflect the triangle in the origin

Now we will reflect the points A', B', and C' in the origin (0, 0). When reflecting a point (x, y) in the origin, both the x and y coordinates change sign. Thus, the reflected points A'', B'', and C'' will be:

- A'' = (-1, 2) (x-coordinate changes from 1 to -1, y-coordinate changes from -2 to 2)
- B'' = (-4, 4) (x-coordinate changes from 4 to -4, y-coordinate changes from -4 to 4)
- C'' = (-3, 7) (x-coordinate changes from 3 to -3, y-coordinate changes from -7 to 7)

### Final Coordinates

The coordinates of the reflected triangle A''B''C'' are:
- A'' = (-1, 2)
- B'' = (-4, 4)
- C'' = (-3, 7)

### Summary of Coordinates
- A'' = (-1, 2)
- B'' = (-4, 4)
- C'' = (-3, 7)

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