A rectangle has two diagonals with the same length. Therefore, use the distance formula to calculate the distance.

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

 \(\displaystyle (7-5)^{2} = 2^{2} = 4\)

\(\displaystyle (-3-2)^{2} = -5^{2} = 25\)

\(\displaystyle \sqrt{4+25}\)

\(\displaystyle \sqrt{29} \approx 5.4\)

\(\displaystyle d \approx 5.4\)

The diameter of the circle would be the longest line that can be drawn within that circle.

Because radius is half of the diameter, the diameter is calculated by multiplying the radius of the circle by two.

If the radius is \(\displaystyle 8cm\),  then the diameter is \(\displaystyle 8m \times 2 = 16m\)

The Pythagorean Theorem states that:

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

With \(\displaystyle a\) and \(\displaystyle b\) representing the measurement of the legs and \(\displaystyle C\) representing the hypotenuse.

\(\displaystyle 10^{2} +b^{2} = 16^{2}\)

\(\displaystyle 100 + b^{2} = 256\)

\(\displaystyle (100-100) + b^{2} = 256-100\)

\(\displaystyle b^{2} = 156\)

\(\displaystyle \sqrt{b^{2}} = \sqrt{156}\)

\(\displaystyle b\approx 12.5ft\)

First, use the Pythagorean Theorem to get the measurement of the other leg.

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

\(\displaystyle 9^{2} + b^{2} =15^{2}\)

\(\displaystyle 81 + b^{2} = 225\)

\(\displaystyle (81-81) + b^{2} = 225-81\)

\(\displaystyle b^{2} = 144\)

\(\displaystyle b = 12cm\)

To get the perimeter of the triangle, add the measurements of each of the three sides.

\(\displaystyle 9 + 15 + 12 = 36cm\)

Use the distance formula

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

\(\displaystyle d=\sqrt{(7-1)^{2} + (9-1)^{2}}\)

\(\displaystyle d=\sqrt{36+64}\)

\(\displaystyle d = \sqrt{100}\)

\(\displaystyle d = 10\)

First, write each portion of the statement in mathematical terms.

\(\displaystyle AC =\frac{1}{2} AB\)

Since AB=80 we will substitute that into the equation.

\(\displaystyle AC =\frac{1}{2} (80)\)

\(\displaystyle AC = 40\)

Now that we know AC we can calculate AD as follows.

\(\displaystyle AD = \frac{1}{2} AC\)

\(\displaystyle AD = \frac{1}{2}(40)\)

\(\displaystyle AD = 20cm\)

Because the radius is the distance from center to any point on a circle, the distance formula is used to find the measurement of the radius.

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

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

\(\displaystyle 1^{2} = 1\times1 = 1\)

\(\displaystyle -3^{2} = -3\times -3 = 9\)

\(\displaystyle d = \sqrt{1+9}\)

\(\displaystyle d =\sqrt{10}\)

Radius is \(\displaystyle \frac{1}{2}\) of the diameter.

If the diameter is \(\displaystyle 12cm\), then the radius is \(\displaystyle 6m.\)

\(\displaystyle \frac{1}{2}\cdot \frac{12}{1}=\frac{12}{2}=\frac{2\cdot6}{2}=6\)

Therefore

\(\displaystyle \frac{1}{3}\) of \(\displaystyle 6cm\) is,

\(\displaystyle \frac{1}{3}\cdot \frac{6}{1}=\frac{6}{3}=\frac{3\cdot 2}{3}=2cm\).

Use the slope formula to solve:

\(\displaystyle m = \frac{y_{2} -y_{1}}{x_{2} - x_{1}}\)

Given the following points.

\(\displaystyle A (-5,-1)=(x_1,y_1)\)

\(\displaystyle B (3,-1)=(x_2,y_2)\)

The slope can be calculated as follows.

\(\displaystyle m = \frac{-1 - (-1)}{3- (-5)}\)

\(\displaystyle m = \frac{0}{8}\)

\(\displaystyle m = 0\)

Because the \(\displaystyle y\) coordinates were the same for points A and B, this would form a horizontal line.  The slope of any horizontal line is \(\displaystyle 0.\)

Find the length of a line from the point \(\displaystyle (9,14)\) to the point \(\displaystyle (-12,36)\).

To find this distance, we need to use distance formula (which is really similar to Pythagorean Theorem)

Distance formula is as follows

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

Where our x's and y's come from our ordered pairs.

So, let's plug and chug

\(\displaystyle d=\sqrt{(9--12)^2+(14-36)^2}\)

Simplify

\(\displaystyle d=\sqrt{(21)^2+(-22)^2}\)

\(\displaystyle d=\sqrt{441+484}=\sqrt{925}\approx30.4\)

So our answer is 30.4