To find the distance from Emma's house to the library, we can use the Pythagorean theorem. The coordinates for Emma's house are (-4, -2) and the library's coordinates are (3, 5).
We start by calculating the horizontal and vertical distances between the two points:
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Horizontal Distance (Δx): This is the difference in the x-coordinates. \[ Δx = x_2 - x_1 = 3 - (-4) = 3 + 4 = 7 \]
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Vertical Distance (Δy): This is the difference in the y-coordinates. \[ Δy = y_2 - y_1 = 5 - (-2) = 5 + 2 = 7 \]
Now we can use the Pythagorean theorem to calculate the distance \(d\) between the two points: \[ d = \sqrt{(Δx)^2 + (Δy)^2} \] Substituting the values we calculated: \[ d = \sqrt{(7)^2 + (7)^2} = \sqrt{49 + 49} = \sqrt{98} \]
To simplify \( \sqrt{98} \): \[ \sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2} \]
Using the approximate value of \( \sqrt{2} \approx 1.414 \): \[ d \approx 7 \times 1.414 \approx 9.898 \]
This gives us the distance approximately. To match this with the options provided, let's square \( 7\sqrt{2} \) to convert to a whole number format instead: \[ d^2 = 98 \]
The distances in the options seem to be numerical values likely aimed at distance in another dimension. The calculations suggest the approach of physical distances in a different context. You may need a clearer option set to determine an appropriate match among A, B, C, D without other numerical context for total distance.
From the calculations made, if we were to match a milestone closer than approximations, the closest answer would likely be the nearest round or alternate integer falling from decomposing number format relative to squarings, not just continuous horizontal and vertical measurements but plausible outcome evaluations of placement apart.
Would you like to narrow further down toward outcomes correlating certain integer scenarios closer?