To find the length of this hypotenuse, we need to use the Pythagorean Theorem:

$\displaystyle c^2=a^2+b^2$, where a and b are the legs and c is the hypotenuse.

Here, c is our missing hypotenuse length, a = 4 ,and b = 14.

Plug these values in and solve for c:

$\displaystyle c^2=4^2+14^2=16+196=212$

$\displaystyle c=\sqrt{212}=\sqrt{4 \cdot 53}=\sqrt{4}\sqrt{53}=2\sqrt{53}$

Use the Pythagorean Theorem: $\displaystyle a^{2}+b^{2}=c^{2}$, where a and b are the legs and c is the hypotenuse.

We know $\displaystyle a$ and $\displaystyle b$, so we can plug them in to solve for c:

$\displaystyle 5^{2}+11^{2}=c^{2}$

$\displaystyle 25+121=c^{2}$

$\displaystyle 146=c^{2}$

$\displaystyle c=\sqrt{146}$

We are really looking for the hypotenuse of a triangle that has legs of 5 miles and 8 miles.

Apply the Pythagorean Theorem:

a² + b² = c²

25 + 64 = c²

89 = c²

c = 9.43 miles

The Pythagorean Theorem states that . This question gives us the values of $\displaystyle a$ and $\displaystyle b$, and asks us to solve for $\displaystyle c$.

Take $\displaystyle 12$ and $\displaystyle 16$ and plug them into the equation as $\displaystyle a$ and $\displaystyle b$:

$\displaystyle 12^{2}+16^{2}=c^{2}$

Now we can start solving for $\displaystyle c$:

$\displaystyle 144+256=c^{2}$

$\displaystyle 400=c^{2}$

$\displaystyle \sqrt{400}=\sqrt{c^{2}}$

$\displaystyle 20=c$

The length of the hypotenuse is $\displaystyle 20$.

By the Pythagorean Theorem, if $\displaystyle c$ is the hypotenuse and $\displaystyle a$ and $\displaystyle b$ are the legs, $\displaystyle c = \sqrt{a^{2}+b^{2}}}$.

Set $\displaystyle a=12$, the known leg, and rewrite the above as:

$\displaystyle c = \sqrt{12^{2}+b^{2}}}$

$\displaystyle c = \sqrt{144+b^{2}}}$

We can now substitute each of the five choices for $\displaystyle b$; the one which yields a whole number for $\displaystyle c$ is the correct answer choice.

$\displaystyle b=4$:

 $\displaystyle c = \sqrt{144+4^{2}}} = \sqrt{144+16}} =\sqrt{160}=12.64...$

$\displaystyle b=12$:

 $\displaystyle c = \sqrt{144+12^{2}}} = \sqrt{144+144}} =\sqrt{288}=16.97...$

$\displaystyle b=15$:

 $\displaystyle c = \sqrt{144+15^{2}}} = \sqrt{144+225}} =\sqrt{369}=19.20...$

$\displaystyle b=16$:

 $\displaystyle c = \sqrt{144+16^{2}}} = \sqrt{144+256}} =\sqrt{400}=20$

$\displaystyle b=20$:

 $\displaystyle c = \sqrt{144+20^{2}}} = \sqrt{144+400}} =\sqrt{544}=23.32...$

The only value of $\displaystyle b$ which yields a whole number for the hypotenuse $\displaystyle c$ is 16, so this is the one we choose.

Divide the shape into a rectangle and a right triangle as indicated below.

Find the hypotenuse of the right triangle with the Pythagorean Theorem, $\displaystyle a^2 + b^2 = c^2$, where $\displaystyle a$ and $\displaystyle b$ are the legs of the triangle and $\displaystyle c$ is its hypotenuse. 

$\displaystyle (8)^2+(6)^2 = c^2$

$\displaystyle 100 = c^2$

$\displaystyle \sqrt{100} = \sqrt{c^2}$

$\displaystyle c =10$

This is our missing length.

Now add the sides of the polygon together to find the perimeter:

$\displaystyle 20 + 10 + 26 + 8 = 64$

Since the lengths of the sides are consecutive integers and the shortest side is $\displaystyle x$, the three sides are $\displaystyle x$, $\displaystyle (x+1)$, and $\displaystyle (x+2)$.

We then use the Pythagorean Theorem:

$\displaystyle \newline a^2+b^2=c^2 \newline x^2+(x+1)^2=(x+2)^2$

To find the distance between points $\displaystyle P$ and $\displaystyle R$, split the square into two 45-45-90 triangles and find the hypotenuse. The side ratio of the 45-45-90 triangle is $\displaystyle s:s:s\sqrt{2}$, so if the sides have a length of 5, the hypotenuse must be $\displaystyle 5\sqrt{2}$.

Plug the values into the Pythagorean Theorem

$\displaystyle a^2 +b^2 = c^2$,

because we are solving for the diagonal we are looking for $\displaystyle c$.

$\displaystyle 8^2 + 15^2 = 289$. 

$\displaystyle \sqrt{289} = 17$.

To find the hypotenuse of a right triangle, use the Pythagorean Theorem and plug the leg values in for $\displaystyle a$ and $\displaystyle b$:
$\displaystyle \\a^2 + b^2 = c^2 \\ \newline 20^2 + 99^2 = c^2 \\ \newline 10,201 = c^2 \\ \\ \sqrt{10,201}=\sqrt{c^2}\\ \newline 101 = c$