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

This question calls for us to use the Pythagorean Theorem. This theorem has a formula of 

\(\displaystyle a^2+b^2=c^2\) 

where a and b are the sides of a right triangle, adjacent to the right angle, and c is the hypotenuse.

Given our information 

\(\displaystyle \\6^2+8^2=c^2\\ =36+64=100\\c=\sqrt{100}=10\).

This question calls for us to use the Pythagorean Theorem. This theorem has a formula of 

\(\displaystyle a^2+b^2=c^2\) 

where a and b are the sides of a right triangle, adjacent to the right angle, and c is the hypotenuse.

Given our information 

\(\displaystyle \\7^2+12^2=c^2\\=49+144=193\\c=\sqrt{193}\).

The Pythagorean Theorem states that for any right triangle with legs \(\displaystyle a\) and \(\displaystyle b\) and hypotenuse \(\displaystyle c\):

\(\displaystyle a^2 + b^2 = c^2\).

Applying this to Marcus's steps, we know that \(\displaystyle 15^2 + 25^2 = c^2\).

Expand:

\(\displaystyle 225 + 625 = 850\)

So, \(\displaystyle c^2 = 850\). To find \(\displaystyle c\), just take the square root:

\(\displaystyle \sqrt{850} = \sqrt{25 \cdot 34} = 5\sqrt{34}\)

So, Marcus travels exactly \(\displaystyle 5\sqrt{34}\textup{ feet}\) back to the start.

This is solving for the hypotenuse of a triangle. Using the Pythagorean Theorem, which says that \(\displaystyle a^2+b^2=c^2\) 

 \(\displaystyle 15^2 + 20^2 = c^2\) 

\(\displaystyle 225+400=c^2\)

\(\displaystyle 625=c^2\)

\(\displaystyle 25=c\)

Susie walks 30 meters north, then 80 meters west, then 30 meters north again. Thus, she walks 60 meters north and 80 meters west. These two directions are 90 degrees away from one another.

At this point, construct a right triangle with one leg that measures 60 meters and a second leg that is 80 meters.

You can save time by using the 3:4:5 common triangle. 60 and 80 are \(\displaystyle 3\cdot 20\) and \(\displaystyle 4\cdot 20\), respectively, making the hypotenuse equal to \(\displaystyle 5\cdot 20=100\).

We can solve for the length of the missing hypotenuse by applying the Pythagorean theorem:

\(\displaystyle a^{2}+b^{2}=c^{2}\)

Substitute the following known values into the formula and solve for the missing hypotenuse: side \(\displaystyle c\).

\(\displaystyle a= 60,\ b= 80,\ c= ?\)

\(\displaystyle (60)^{2}+(80)^{2}=c^{2}\)

 \(\displaystyle 3600+6400=c^{2}\)

\(\displaystyle 10,000=c^{2}\)

\(\displaystyle c=100\)

Susie will walk 100 meters to reach her house.

First, define the sides of the triangle. Because the side lengths are consecutive odd numbers, if we define the shortest side will be as \(\displaystyle x\), the next side will be defined as \(\displaystyle x+2\), and the longest side will be defined as \(\displaystyle x+4\). We can then find the perimeter of a triangle using the following formula:

\(\displaystyle \text{Perimeter}=\text{side}+\text{side}+\text{side}\)

Substitute in the known values and variables.

\(\displaystyle \text{Perimeter}=x+(x+2)+(x+4)\)

\(\displaystyle 57=3x+6\)

Subtract 6 from both sides of the equation.

\(\displaystyle 57-6=3x+6-6\)

\(\displaystyle 51=3x\)

Divide both sides of the equation by 3. 

\(\displaystyle \frac{51}{3}=\frac{3x}{3}\)

Solve.

\(\displaystyle x=17\)

This is not the answer; we need to find the length of the longest side, or \(\displaystyle x+4\). 

\(\displaystyle \text{Longest side}=x+4\)

Substitute in the calculated value for \(\displaystyle x\) and solve.

\(\displaystyle \text{Longest side}=17+4\)

\(\displaystyle \text{Longest side}=21\)

The longest side of the triangle is 21 centimeters long.

\(\displaystyle 6, 7, 12\) cannot be the side lengths of a right triangle. \(\displaystyle 6^2+ 7^2\) does not equal \(\displaystyle 12^2\). Also, special right triangle \(\displaystyle 3-4-5, 5-12-13,\) and \(\displaystyle 45-45-90\) rules can eliminate all the other choices.

Using Pythagorean Theorem, we can solve for the length of leg x:

x² + 6² = 10²

Now we solve for x:

x² + 36 = 100

x² = 100 – 36

x² = 64

x = 8

Also note that this is proportionally a 3/4/5 right triangle, which is very common. Always look out for a side-to-hypoteneuse ratio of 3/5 or 4/5, or a side-to-side ratio of 3/4, in any right triangle, so that you may solve such triangles rapidly.

Using the pythagorean theorem, 8²=7²+x². Solving for x yields the square root of 15, which is 3.9