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b) Imagine you are an architect designing a room. You have three key points that represent the location of a support column in a triangular room: [tex]\( X(-3, 5) \)[/tex], [tex]\( Y(4, 5) \)[/tex], and [tex]\( Z(-3, 1) \)[/tex].

1. Plot and join these three points on graph paper to visualize the room's layout.

2. If you decide to add a mirror wall along the X-axis to create an illusion in the same room, find the coordinates of the new positions of the support columns after the mirror installation and visualize them on graph paper.

(Note: This task is related to question 19(b) from the previous set.)



Answer :

GinnyAnswer
Certainly! Let's proceed step-by-step with the given information.

### Step 1: Plotting the Original Coordinates

We start by plotting the original coordinates of the support columns on graph paper.

1. Point X: [tex]\((-3, 5)\)[/tex]
2. Point Y: [tex]\((4, 5)\)[/tex]
3. Point Z: [tex]\((-3, 1)\)[/tex]

### Step 2: Drawing the Triangle

Once the points are plotted, you can connect them to observe the triangular room layout.

- Draw a line from point X [tex]\((-3, 5)\)[/tex] to point Y [tex]\((4, 5)\)[/tex].
- Draw a line from point Y [tex]\((4, 5)\)[/tex] to point Z [tex]\((-3, 1)\)[/tex].
- Draw a line from point Z [tex]\((-3, 1)\)[/tex] back to point X [tex]\((-3, 5)\)[/tex].

This forms a triangle representing the layout of the room with the given support columns.

### Step 3: Reflecting Points Across the X-axis

To add a mirror wall along the X-axis and find the coordinates of the new positions of the support columns, we simply reflect the y-coordinates of each point across the X-axis. This means negating the y-coordinates.

1. Original Point X: [tex]\((-3, 5)\)[/tex]
- Reflected Point X: [tex]\((-3, -5)\)[/tex]
2. Original Point Y: [tex]\((4, 5)\)[/tex]
- Reflected Point Y: [tex]\((4, -5)\)[/tex]
3. Original Point Z: [tex]\((-3, 1)\)[/tex]
- Reflected Point Z: [tex]\((-3, -1)\)[/tex]

### Step 4: Plotting the Reflected Coordinates

Next, we plot the reflected coordinates on the same graph paper.

1. Reflected Point X: [tex]\((-3, -5)\)[/tex]
2. Reflected Point Y: [tex]\((4, -5)\)[/tex]
3. Reflected Point Z: [tex]\((-3, -1)\)[/tex]

### Step 5: Drawing the Reflected Triangle

Finally, connect the reflected points to visualize the new positions.

- Draw a line from reflected point X [tex]\((-3, -5)\)[/tex] to reflected point Y [tex]\((4, -5)\)[/tex].
- Draw a line from reflected point Y [tex]\((4, -5)\)[/tex] to reflected point Z [tex]\((-3, -1)\)[/tex].
- Draw a line from reflected point Z [tex]\((-3, -1)\)[/tex] back to reflected point X [tex]\((-3, -5)\)[/tex].

Now you have both the original and the reflected triangles on the graph paper, showing how the mirror wall will create the illusion in the same room.

### Summary of New Coordinates

The new coordinates of the support columns after the mirror installation along the X-axis are:
- Reflected Point X: [tex]\((-3, -5)\)[/tex]
- Reflected Point Y: [tex]\((4, -5)\)[/tex]
- Reflected Point Z: [tex]\((-3, -1)\)[/tex]

These steps help us visualize the room's layout and the effect of adding a mirror wall along the X-axis.

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