Question

find the missing side of the triangle. write your answer in simplified radical form.
11.
$sqrt{6}$ ft
$2\sqrt{2}$ ft
$x$
12.
$sqrt{7}$ m
$2\sqrt{3}$ m
$x$
13.
$2\sqrt{2}$ m
$sqrt{10}$ m
$x$
14.
$2\sqrt{2}$ mi
$sqrt{11}$ mi
$x$
use the distance formula to find the distance between the two points on the graph. leave your answer in simplified radical form.
15.
16.

Explanation:

Step1: Apply Pythagorean theorem (hypotenuse)

$x = \sqrt{(\sqrt{6})^2 + (2\sqrt{2})^2} = \sqrt{6 + 8} = \sqrt{14}$

Step2: Apply Pythagorean theorem (hypotenuse)

$x = \sqrt{(\sqrt{7})^2 + (2\sqrt{3})^2} = \sqrt{7 + 12} = \sqrt{19}$

Step3: Apply Pythagorean theorem (leg)

$x = \sqrt{(\sqrt{10})^2 - (2\sqrt{2})^2} = \sqrt{10 - 8} = \sqrt{2}$

Step4: Apply Pythagorean theorem (leg)

$x = \sqrt{(\sqrt{11})^2 - (2\sqrt{2})^2} = \sqrt{11 - 8} = \sqrt{3}$

Step5: Identify points, use distance formula

Points: $\boldsymbol{a=(2,4)}$, $\boldsymbol{b=(-3,-2)}$.
$d = \sqrt{(2 - (-3))^2 + (4 - (-2))^2} = \sqrt{5^2 + 6^2} = \sqrt{25 + 36} = \sqrt{61}$

Step6: Identify points, use distance formula

Points: $(-3,-4)$, $(3,4)$.
$d = \sqrt{(3 - (-3))^2 + (4 - (-4))^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$

Answer:

1. $\boldsymbol{\sqrt{14}}$ ft
2. $\boldsymbol{\sqrt{19}}$ m
3. $\boldsymbol{\sqrt{2}}$ m
4. $\boldsymbol{\sqrt{3}}$ mi
5. $\boldsymbol{\sqrt{61}}$ units
6. $\boldsymbol{10}$ units