Question

biking rhea is using a coordinate grid to map out how far she biked from each place in town. points x, y, and z are all locations in rheas town.

Explanation:

Step1: Identify coordinates

Let \(X=(2,0)\), \(Y=(6, - 3)\), \(Z=(2,-9)\).

Step2: Use distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

For distance between \(X\) and \(Y\):
\[

$$\begin{align*} d_{XY}&=\sqrt{(6 - 2)^2+(-3-0)^2}\\ &=\sqrt{4^2+(-3)^2}\\ &=\sqrt{16 + 9}\\ &=\sqrt{25}\\ &=5 \end{align*}$$

\]

Step3: For distance between \(X\) and \(Z\)

\[

$$\begin{align*} d_{XZ}&=\sqrt{(2 - 2)^2+(-9 - 0)^2}\\ &=\sqrt{0+(-9)^2}\\ &=\sqrt{81}\\ &=9 \end{align*}$$

\]

Step4: For distance between \(Y\) and \(Z\)

\[

$$\begin{align*} d_{YZ}&=\sqrt{(2 - 6)^2+(-9+3)^2}\\ &=\sqrt{(-4)^2+(-6)^2}\\ &=\sqrt{16 + 36}\\ &=\sqrt{52}\\ &=2\sqrt{13}\approx 7.21 \end{align*}$$

\]

Answer:

Distance between \(X\) and \(Y\) is \(5\), distance between \(X\) and \(Z\) is \(9\), distance between \(Y\) and \(Z\) is \(2\sqrt{13}\approx7.21\)