To solve this problem, we can visualize the scenario using a right triangle.
- Natalie’s house (let’s denote it as point \(N\)) is at the origin \((0, 0)\).
- Chloe's house (point \(C\)) is directly north of Natalie’s house, which means it has coordinates \((0, y)\) for some distance \(y\).
- 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.