Question

You have become an in sensation after your video received over 2 million views in one week this made you $1 million. You call a builder and ask them to build you a house twice the size of your current house. The new coordinations of your house are (- 4, 0), (-4, 4), (- 2, 6), (0,4) and (0,0)

Answer 1

To find the coordinates of the new house that is twice the size of your current house, we first need to understand the dimensions and shape of the original house based on the provided coordinates.

The coordinates you've provided are:

1. (-4, 0)
2. (-4, 4)
3. (-2, 6)
4. (0, 4)
5. (0, 0)

We can visualize these points on a Cartesian plane. It seems to form a shape, and we will determine the centroid or center to appropriately scale the dimensions for the new house.

1. **Finding the centroid (center point)**:
The centroid \((C_x, C_y)\) can be found by averaging the x-coordinates and the y-coordinates:

\[
C_x = \frac{x_1 + x_2 + x_3 + x_4 + x_5}{5} = \frac{-4 + -4 + -2 + 0 + 0}{5} = \frac{-10}{5} = -2
\]

\[
C_y = \frac{y_1 + y_2 + y_3 + y_4 + y_5}{5} = \frac{0 + 4 + 6 + 4 + 0}{5} = \frac{14}{5} = 2.8
\]

The centroid of the shape is at the point \((-2, 2.8)\).

2. **Scaling the coordinates**:
To build a house twice the size, we need to scale the dimensions while keeping the centroid the same. This involves moving each point away from the centroid by the same ratio.

Each original point \((x, y)\) will be transformed to \((x', y')\) using the scaling factor (for size):

\[
x' = C_x + 2 \cdot (x - C_x)
\]
\[
y' = C_y + 2 \cdot (y - C_y)
\]

Applying this to each point:

1. **For (-4, 0)**:
\[
x' = -2 + 2 \cdot (-4 + 2) = -2 + 2 \cdot (-2) = -2 - 4 = -6
\]
\[
y' = 2.8 + 2 \cdot (0 - 2.8) = 2.8 - 5.6 = -2.8
\]
Result: \((-6, -2.8)\)

2. **For (-4, 4)**:
\[
x' = -2 + 2 \cdot (-4 + 2) = -6
\]
\[
y' = 2.8 + 2 \cdot (4 - 2.8) = 2.8 + 2.4 = 5.2
\]
Result: \((-6, 5.2)\)

3. **For (-2, 6)**:
\[
x' = -2 + 2 \cdot (-2 + 2) = -2
\]
\[
y' = 2.8 + 2 \cdot (6 - 2.8) = 2.8 + 6.4 = 9.2
\]
Result: \((-2, 9.2)\)

4. **For (0, 4)**:
\[
x' = -2 + 2 \cdot (0 + 2) = -2 + 4 = 2
\]
\[
y' = 2.8 + 2 \cdot (4 - 2.8) = 5.2
\]
Result: \((2, 5.2)\)

5. **For (0, 0)**:
\[
x' = -2 + 2 \cdot (0 + 2) = 2
\]
\[
y' = 2.8 + 2 \cdot (0 - 2.8) = -2.8
\]
Result: \((2, -2.8)\)

3. **Final coordinates**:
The new coordinates of your house that is twice the size of your current house are:

1. \((-6, -2.8)\)
2. \((-6, 5.2)\)
3. \((-2, 9.2)\)
4. \((2, 5.2)\)
5. \((2, -2.8)\)

These points represent the vertices of the new house based on scaling from the centroid of the original house.