GED Math - Distance Formula

Provide your answer in its most simplified form.

Find the distance between the two following points:

$\sqrt{137}$

$\sqrt{55}$

$\sqrt{25}$

Explanation

We must use the distance formula to solve this problem:

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

Plug in your x and y values:

$d = \sqrt{(2 - 5)^{2} + (2 - 6)^{2}}$

Combine like terms:

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

Continue with your order of operations

$d = \sqrt{9 + 16}$

$d = \sqrt{25}$

Simplify to get:

Use distance formula to find the distance between the following two points.

$5\sqrt{37}$

$37\sqrt{5}$

$5\sqrt{39}$

Explanation

Use distance formula to find the distance between the following two points.

Distance formula is as follows:

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

Note that it doesn't matter which point is "1" and which point is "2" just so long as we remain consistent.

So, let's plug and chug.

$d=\sqrt{(43-22)^2+(54-76)^2}=\sqrt{21^2+(-22)^2}=\sqrt{441+484}=\sqrt{925}$

$d=\sqrt{925}=\sqrt{25}\sqrt{37}=5\sqrt{37}$

So, our answer is

$5\sqrt{37}$

A triangle on a coordinate plane has the following vertices: . What is the perimeter of the triangle?

Explanation

Since we are asked to find the perimeter of the triangle, we will need to use the distance formula to find the length of each side. Recall the distance formula:

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

Start by finding the distance between the points $(5, 5)\text{ and }(10, 2)$ :

$\text{Distance}=\sqrt{(2-5)^2+(10-5)^2}=\sqrt{9+25}=\sqrt{34}$

Next, find the distance between $(10, 2)\text{ and }(8, -2)$ .

$\text{Distance}=\sqrt{(-2-2)^2+(8-10)^2}=\sqrt{16+4}=\sqrt{20}$

Then, find the distance between $(5, 5)\text{ and }(8, -2)$ .

$\text{Distance}=\sqrt{(-2-5)^2+(8-5)^2}=\sqrt{49+9}=\sqrt{58}$

Finally, add up the lengths of each side to find the perimeter of the triangle.

$\text{Perimeter}=\sqrt{54}+\sqrt{20}+\sqrt{58}=17.92$

Find the distance between the points and .

Explanation

Find the distance between the points and .

To find the distance between two points, we will use distance formula (clever name). Distance formula can be thought of as a modified Pythagorean Theorem. What distance formula does is essentially treats our two points as the ends of a hypotenuse on a right triangle, then uses the two side lengths to find the hypotenuse.

Distance formula:

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

Pythagorean Theorem

If the connection isn't clear, don't worry, we can still solve for distance.

$D=\sqrt{(413-9)^2+(65-17)^2}$

$D=\sqrt{(404)^2+(48)^2}$

$D=\sqrt{165520}$

$D=406.84\approx 407$

So our answer is 407

Find the length of the line connecting the following points.

Explanation

Find the length of the line connecting the following points.

To find the length of a line, use distance formula.

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

What we are really doing is making a right triangle and using Pythagorean Theorem to find the hypotenuse.

Let's plug in our points and find the distance!

$d=\sqrt{(123-47)^2+(99-19)^2}$

$d=\sqrt{(76)^2+(80)^2}$

$d=\sqrt{(5776+6400}=\sqrt{12176}$

$\sqrt{12176}\approx 110$

So our answer is 110

What is the distance between the points and ?

$\sqrt{181}$

$6\sqrt{3}$

$\sqrt{146}$

$12\sqrt{2}$

Explanation

Recall the distance formula:

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

Plug in the given points to find the distance between them.

$\text{Distance}=\sqrt{(5-(-5))^2+(1-10)^2}=\sqrt{100+81}=\sqrt{181}$

The distance between those points is $\sqrt{181}$ .

Provide your answer in its most simplified form.

Find the distance between these two points:

$\sqrt{313}$

$\sqrt{65}$

$\sqrt{193}$

$\sqrt{-313}$

$\sqrt{33}$

Explanation

For this problem we must use the distance formula:

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

Plug in your x and y values:

$d = \sqrt{((-10) - 3)^{2} + ((-4) - 8)^{2}}$

Combine like terms:

$d = \sqrt{(-13)^{2} + (-12)^{2}}$

Continue your order of operations:

$d = \sqrt{169 +144}$

$d = \sqrt{313}$

This cannot be simplified so you are left with the correct answer.

What is the distance between the points and ?

$5\sqrt{13}$

$25\sqrt{3}$

$10\sqrt{7}$

$3\sqrt{5}$

Explanation

Remember that you can consider your two points as:

and

From this, remember that the distance formula is:

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

Now, for your data, this is:

$\sqrt{(22-4)^2+(6-5)^2}$

$\sqrt{(18)^2+(1)^2}=\sqrt{324+1}=\sqrt{325}$

You can simplify this value a little. Identify the prime factors of and move any number that appears in a pair of factors from the interior to the exterior of the square root symbol: