To determine the distance between Kelvin's house and Mitch's house, we first need to find the coordinates of Mitch's house. We know that the store at (1, 0) is halfway between Kelvin's house at (3, 4) and Mitch's house.
Let \( M = (x, y) \) be the coordinates of Mitch's house. The midpoint formula gives us the coordinates of the midpoint as:
\[ \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Applying this to our situation:
\[ \left( \frac{3 + x}{2}, \frac{4 + y}{2} \right) = (1, 0) \]
This gives us the following two equations:
- \(\frac{3 + x}{2} = 1\)
- \(\frac{4 + y}{2} = 0\)
Next, we solve these equations:
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For the first equation: \[ 3 + x = 2 \ x = 2 - 3 \ x = -1 \]
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For the second equation: \[ 4 + y = 0 \ y = -4 \]
So, Mitch's house is located at \( (-1, -4) \).
Now we can calculate the distance between Kelvin's house (3, 4) and Mitch's house (-1, -4) using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting in the coordinates of Kelvin's and Mitch's house:
\[ d = \sqrt{((-1) - 3)^2 + ((-4) - 4)^2} \ d = \sqrt{(-4)^2 + (-8)^2} \ d = \sqrt{16 + 64} \ d = \sqrt{80} \ d = 4\sqrt{5} \]
Calculating \( 4\sqrt{5} \) to the nearest tenth:
\[ \sqrt{5} \approx 2.236 \ 4\sqrt{5} \approx 4 \times 2.236 \approx 8.944 \]
Rounding this to the nearest tenth gives:
\[ \text{Distance} \approx 8.9 \]
Thus, the distance between Kelvin's house and Mitch's house is approximately 8.9.