find the distance between the points. round to two decimal places if necessary.

Explanation:

Step1: Identify the points

The points are $(-4, 6)$, $(-7, 12)$, and $(5, -9)$. We calculate distances between each pair, then confirm the full set.

Step2: Distance formula setup

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

Step3: Distance between $(-4,6)$ and $(-7,12)$

$$ \begin{align*} d_1&=\sqrt{(-7-(-4))^2+(12-6)^2}\ &=\sqrt{(-3)^2+6^2}\ &=\sqrt{9+36}\ &=\sqrt{45}\approx6.71 \end{align*} $$

Step4: Distance between $(-4,6)$ and $(5,-9)$

$$ \begin{align*} d_2&=\sqrt{(5-(-4))^2+(-9-6)^2}\ &=\sqrt{9^2+(-15)^2}\ &=\sqrt{81+225}\ &=\sqrt{306}\approx17.49 \end{align*} $$

Step5: Distance between $(-7,12)$ and $(5,-9)$

$$ \begin{align*} d_3&=\sqrt{(5-(-7))^2+(-9-12)^2}\ &=\sqrt{12^2+(-21)^2}\ &=\sqrt{144+441}\ &=\sqrt{585}\approx24.19 \end{align*} $$

Answer:

Distance between $(-4,6)$ and $(-7,12)$: $\approx6.71$ Distance between $(-4,6)$ and $(5,-9)$: $\approx17.49$ Distance between $(-7,12)$ and $(5,-9)$: $\approx24.19$