To find the coordinates of the cabin using the midpoint formula, we'll define the coordinates of the church (C) and the cabin (K) as follows:
Let:
- Church (C) be at \((x_1, y_1)\)
- Cabin (K) be at \((x_2, y_2)\)
- House (H, the midpoint) be at \((x_m, y_m)\)
The midpoint formula states that: \[ (x_m, y_m) = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \]
Finding the x-coordinate of the cabin
We can rearrange the formula to solve for \(x_2\): \[ x_m = \frac{x_1+x_2}{2} \] To isolate \(x_2\), we multiply both sides by 2: \[ 2x_m = x_1 + x_2 \] Now, solving for \(x_2\): \[ x_2 = 2x_m - x_1 \]
Substitution: If we know the coordinates of the Church (C) and House (H) (midpoint), we can substitute the values:
- For example, if \(C\) = (3, 4) and \(H\) = (5, 6), the calculations would be:
- \(x_1 = 3\)
- \(x_m = 5\)
Now plugging in: \[ x_2 = 2(5) - 3 = 10 - 3 = 7 \]
Finding the y-coordinate of the cabin
Using the same process for the y-coordinates: \[ y_m = \frac{y_1 + y_2}{2} \] We isolate \(y_2\): \[ y_2 = 2y_m - y_1 \]
Substitution: If we know the corresponding \(y\) values:
- Assuming \(C = (3, 4)\) and \(H = (5, 6)\),
- \(y_1 = 4\)
- \(y_m = 6\)
Now plugging in: \[ y_2 = 2(6) - 4 = 12 - 4 = 8 \]
Final Coordinates of the Cabin
Now we have:
- \(x_2 = 7\)
- \(y_2 = 8\)
Thus, the final coordinates of the cabin are \((7, 8)\).
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