Question

from From her home, Natalie would have to walk due north to get to her friend Chloe's house and due east to get to her friend Jada's house. It is 8 kilometers from Natalie's house to Jada's house and a straight-line distance of 10 kilometers from Chloe's house to Jada's house. How far is Natalie's house from Chloe's house? kilometers

Answer 1

To solve this problem, we can visualize the scenario using a right triangle.

1. Natalie’s house (let’s denote it as point \(N\)) is at the origin \((0, 0)\).
2. Chloe's house (point \(C\)) is directly north of Natalie’s house, which means it has coordinates \((0, y)\) for some distance \(y\).
3. Jada's house (point \(J\)) is directly east of Natalie’s house, which means it has coordinates \((x, 0)\) for some distance \(x\).

Given:
- The distance from Natalie's house to Jada's house is \(x = 8\) kilometers.
- The straight-line distance from Chloe's house to Jada's house is \(10\) kilometers. The coordinates of Chloe and Jada give us the distance as follows:

Using the distance formula:
\[
\text{Distance between } C \text{ and } J = \sqrt{(x - 0)^2 + (0 - y)^2} = 10
\]
Substituting \(x = 8\):
\[
\sqrt{(8 - 0)^2 + (0 - y)^2} = 10
\]
\[
\sqrt{8^2 + y^2} = 10
\]
\[
\sqrt{64 + y^2} = 10
\]
Now, square both sides:
\[
64 + y^2 = 100
\]
\[
y^2 = 100 - 64
\]
\[
y^2 = 36
\]
\[
y = 6
\]

Now that we have \(y\), which is the distance from Natalie's house to Chloe's house, we can conclude that:

**The distance from Natalie's house to Chloe's house is \(6\) kilometers.**