Question

14 multiple choice 5 points what is the distance between the two points in simplest radical form? p(2,8) and b(1,3)
options: 2√13, 6, √26, √130
what is the perimeter of triangle abc? round the answer to the nearest tenth, if necessary.
options: 10 units, 11 units, 12 units, 13 units
a(-1,3), c(3,-3), b(7,0)
ab = √((-1 - 3)²+(3 - 0)²)
=√((-4)²+(-3)²)

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $P(2,8)$ and $B(1,3)$, let $(x_1,y_1)=(2,8)$ and $(x_2,y_2)=(1,3)$.

Step2: Substitute values into formula

$d=\sqrt{(1 - 2)^2+(3 - 8)^2}=\sqrt{(-1)^2+(-5)^2}=\sqrt{1 + 25}=\sqrt{26}$.

For the perimeter of $\triangle ABC$ with $A(-1,3)$, $B(3,0)$ and $C(3, - 3)$:

Step3: Calculate length of $AB$

$AB=\sqrt{(-1 - 3)^2+(3-0)^2}=\sqrt{(-4)^2+3^2}=\sqrt{16 + 9}=\sqrt{25}=5$.

Step4: Calculate length of $BC$

The $x$-coordinates of $B(3,0)$ and $C(3,-3)$ are the same. So, $BC=\vert0-(-3)\vert = 3$.

Step5: Calculate length of $AC$

$AC=\sqrt{(-1 - 3)^2+(3-(-3))^2}=\sqrt{(-4)^2+6^2}=\sqrt{16 + 36}=\sqrt{52}=2\sqrt{13}\approx 7.2$.

Step6: Calculate perimeter

Perimeter $P=AB + BC+AC=5 + 3+7.2 = 15.2\approx15$ (rounded to nearest whole - number). But if we consider the non - rounded values and add $\sqrt{25}+3 + 2\sqrt{13}=5 + 3+2\sqrt{13}=8 + 2\sqrt{13}$.

Answer:

For the distance between $P(2,8)$ and $B(1,3)$: $\sqrt{26}$; For the perimeter of $\triangle ABC$: approximately $15$ units.