By the Segment Addition Postulate,

$\displaystyle AB + BC + CD + DE = AE$

$\displaystyle x + (x+4) + (2x+1) + (x-6) = 80$

$\displaystyle 5x-1= 80$

$\displaystyle 5x-1+1 = 80+1$

$\displaystyle 5x = 81$

$\displaystyle 5x \div 5 = 81 \div 5$

$\displaystyle x = 16.2$

$\displaystyle AB = x = 16.2$

In geometry, a right angle triangle can occur with the ratio of $\displaystyle 3:4:5$ in which 3 and 4 are each leg lengths, and 5 is the hypotenuse. 

When you know the length of two sides of a right angle triangle like this, you can calculate the third side using this ratio.

Here, the ratio is:

$\displaystyle 6:x:10$

This is double the  $\displaystyle 3:4:5$ ratio. Therefore, we should multiply 4 by 2 in order to solve for the missing leg, which would be a value of 8 feet. 

Another way to solve is to use the Pythagorean Theorem: $\displaystyle a^2+b^2=c^2$.

We know that one leg is 6 feet and the hypotenuse is 10 feet.

$\displaystyle (6)^2+b^2=(10)^2$

$\displaystyle 36+b^2=100$

$\displaystyle b^2=64$

$\displaystyle b=\sqrt{64}=8$

If the radius is equal to 6 inches, then the diameter will be double that value, or 12 inches. One-third of 12 is 4, which is therefore the correct answer. 

Since we are dealing with a right triangle, we can use the Pythagorean Theorem:

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

where $\displaystyle a$ and $\displaystyle b$ are leg lengths of $\displaystyle 5$ and $\displaystyle 8$, respectively, and $\displaystyle c$ is the length of the hypotenuse.

Substituting values into the Theorem:

$\displaystyle 5^{2}+8^{2}=c^{2}$

$\displaystyle 25+64=c^{2}$

$\displaystyle 89=c^{2}$

$\displaystyle \sqrt{89}=c$

A line that is bisected is split into two segments of equal length. Therefore, if line $\displaystyle AB$ is bisected at point $\displaystyle C$, 

$\displaystyle AC=\frac{1}{2}AB=30\: cm$.

Consequently, bisecting line segment $\displaystyle AC$ at point $\displaystyle D$:

$\displaystyle AD=\frac{1}{2}AC=\frac{1}{2}(30)=15\:cm$

By the Segment Addition Postulate,

$\displaystyle AB + BC + CD + DE = AE$

$\displaystyle x +( x + 3) +( 2x) +( x-6) = 40$

$\displaystyle 5x-3 = 40$

$\displaystyle 5x-3 +3 = 40 + 3$

$\displaystyle 5x = 43$

$\displaystyle 5x \div 5 = 43 \div 5$

$\displaystyle x = 8.6$

$\displaystyle AB = x = 8.6$

The length of a line segment can be determined using the distance formula:

$\displaystyle D=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$

$\displaystyle D=\sqrt{(3-6)^2+(5-1)^2}$

$\displaystyle D=\sqrt{-3^2+4^2}$

$\displaystyle D=\sqrt{9+16}=\sqrt{25}=5$

To find the length of this line, you can subtract $\displaystyle 10-1$ to get $\displaystyle 9$. Since the y-coordinates are the same, you don't have to take any vertical direction into account. Therefore, you only look at the x-coordinates!

We can use the distance formula:

$\displaystyle D=\sqrt{(x_{1}-x_{2})^2+(y_{1}-y_{2})^2}$

$\displaystyle =\sqrt{(-3-5)^2+(5-4)^2}$

$\displaystyle =\sqrt{64+1}$

$\displaystyle =\sqrt{65}\approx 8.06$

The radius is the distance from the center of the circle to anypoint on the circle. So we can use the distance formula in order to find the radius of the circle:

$\displaystyle D=\sqrt{(x_{1}-x_{2})^2+(y_{1}-y_{2})^2}$

$\displaystyle =\sqrt{(4-3)^2+(4-8)^2}$

$\displaystyle =\sqrt{1+16}$

$\displaystyle =\sqrt{17}\approx 4.12$