Question

use the distance formula to determine the measures of the sides of $\triangle abc$.
the coordinates of $\triangle abc$ are $a(0,3)$, $b(4,-2)$, and $c(-4,-2)$.
enter the coordinates of the vertices into the correct positions in the distance formula.
$ab = \sqrt{\square - 4)^2 + \square - (-2)^2}$ or $\square$ units
$ac = \sqrt{(0 - \square))^2 + 3 - (\square)^2}$ or $\square$ units
$bc = \sqrt{4 - (\square)^2 + -2 - (\square)^2}$ or $\square$ units

Explanation:

Step1: Recall the Distance Formula

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Step2: Calculate \(AB\)

For points \(A(0, 3)\) and \(B(4, -2)\), \(x_1 = 0\), \(y_1 = 3\), \(x_2 = 4\), \(y_2 = -2\).
Substitute into the distance formula: \(AB=\sqrt{(4 - 0)^2+[-2 - 3]^2}=\sqrt{4^2+(-5)^2}=\sqrt{16 + 25}=\sqrt{41}\) units.
So the first blank in \(AB\) is \(0\) (since \(x_1 = 0\)), the second blank is \(3\) (since \(y_1 = 3\)), and the length is \(\sqrt{41}\).

Step3: Calculate \(AC\)

For points \(A(0, 3)\) and \(C(-4, -2)\), \(x_1 = 0\), \(y_1 = 3\), \(x_2 = -4\), \(y_2 = -2\).
Substitute into the distance formula: \(AC=\sqrt{(-4 - 0)^2+[-2 - 3]^2}=\sqrt{(-4)^2+(-5)^2}=\sqrt{16 + 25}=\sqrt{41}\) units.
So the first blank in \(AC\) is \(-4\) (since \(x_2=-4\)), the second blank is \(-2\) (since \(y_2 = -2\)), and the length is \(\sqrt{41}\).

Step4: Calculate \(BC\)

For points \(B(4, -2)\) and \(C(-4, -2)\), \(x_1 = 4\), \(y_1 = -2\), \(x_2 = -4\), \(y_2 = -2\).
Substitute into the distance formula: \(BC=\sqrt{(-4 - 4)^2+[-2 - (-2)]^2}=\sqrt{(-8)^2+0^2}=\sqrt{64}=8\) units.
So the first blank in \(BC\) is \(-4\) (since \(x_2=-4\)), the second blank is \(-2\) (since \(y_1 = -2\)), and the length is \(8\).

Answer:

• For \(AB\): First blank \(0\), second blank \(3\), length \(\boldsymbol{\sqrt{41}}\) units.
• For \(AC\): First blank \(-4\), second blank \(-2\), length \(\boldsymbol{\sqrt{41}}\) units.
• For \(BC\): First blank \(-4\), second blank \(-2\), length \(\boldsymbol{8}\) units.