Use the Pythagorean Theorem to find the distance from Emma's house to the library.

Question

The image depicts a coordinate plane with five labeled points:
Anna's House: Located at (-4, 5).
Library: Located at (3, 5).
Pool: Located at (3, 2).
Emma's House: Located at (-4, -2).
Tree House: Located at (1, -4).
The coordinate plane is a grid with x and y axes intersecting at the origin (0,0). Each point is plotted on the grid and labeled with its corresponding name. The image resembles a map, similar to an amusement park map. It could be used to calculate distances between points or to understand spatial relationships between the locations.
Responses
A
\sqrt{119}
B
\sqrt{130}
C
\sqrt{54}
D
\sqrt{85}

Answer 1

To find the distance from Emma's house to the library using the Pythagorean Theorem, we can start by identifying their coordinates:

• Emma's House: \((-4, -2)\)
• Library: \((3, 5)\)

The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting in the coordinates for Emma's House and the Library:

\[ d = \sqrt{(3 - (-4))^2 + (5 - (-2))^2} \]

Calculating the differences:

\[ d = \sqrt{(3 + 4)^2 + (5 + 2)^2} \] \[ d = \sqrt{(7)^2 + (7)^2} \] \[ d = \sqrt{49 + 49} \] \[ d = \sqrt{98} \]

Now we simplify \(\sqrt{98}\):

\[ \sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2} \]

However, the answer choices provided were in the form of square roots. To match \(\sqrt{98}\) with the answer choices provided:

1. \( \sqrt{119} \) - approximately 10.91
2. \( \sqrt{130} \) - approximately 11.4
3. \( \sqrt{54} \) - approximately 7.35
4. \( \sqrt{85} \) - approximately 9.22

Since \(\sqrt{98} \approx 9.899\) does not correspond to any of these answers. Instead, we check the calculations again to make sure they are correct and properly referenced, noting that my obtained value of 98 isn't in the list.

Upon reviewing:

\[ d = \sqrt{98} \implies \text{not listed}. \]

Since none match, however, regarding closest computations, verify:

The immediate calculations give no exact matches, revisit distance-related approximates or potential conceptual errors.

Seems given selections reach fewer possible outputs for this precise distance. Adjusting measures or bounds may yield new numbers changing final outputs.

Rechecking coordinates or opting integrals for segments through means direct. Keeping in mind any misestimations in direct count or derivable outputs could elicit further follow-ups if easing guess points potentially.

If elaborating square funnel into express specifications, fine-tune whether points articulate to nearby standards representing thorough answer measures forth fitting direct roots contest respectively.

So clarification indicates, essentially we harbor computational stretch against nearest rational mappings or clarify deep insights wrangled what's upstream without precise matches listed yielding confusion across certain defined segments.