How to find the length of a line with distance formula - Math

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Question

Find the length of the line segment from the origin to the point (3, 4).

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Answer

Here, we need to use the distance formula between the two points (0, 0) and (3, 4).

$d=$\sqrt{(x_2-x_$1)^2$+(y_2-y_$1)^2$}$$

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Question

I have two points, (–8,3) and (6,–1). If I want to connect those two points with a line segment, how long would that line segment need to be?

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Answer

To determine how long the line needs to be to connect those two points, we need to use the distance formula, shown below.

$d=$\sqrt{(x_2-x_$1)^2$+(y_2-y_$1)^2$}$$

The two points are and . In our case, the points are (–8, 3) and (6, –1).

$d=$\sqrt{196+14}$$

$d=$\sqrt{212}$$

$d=$\sqrt{4(53)}$$

$d=2$\sqrt{53}$$

So in order to connect the two points, the length of the line needs to have $d=2$\sqrt{53}$$ .

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Question

What is the distance between the points and ?

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Answer

To solve problems like this, we simply need to use the distance formula, $d = $$\sqrt{(x_{2}$$-x_${1})^${2}$+(y_{2}$-y_${1})^{2}$}$ . Plugging in the and values from our points yields $d = $$\sqrt{(10-7)^{2}$$$+(4-8)^{2}$}$ , or $d = $$\sqrt{(3)^{2}$$$+(-4)^{2}$}$ . Solving this radical gives us a value of $d = $\sqrt{25}$$ , or 5.

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Question

Find the length of the line segment with endpoints at and .

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Answer

Use the distance formula, with :

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

$= $\sqrt{ \left (2 -7\right ) ^{2}$ + \left(9 -6 \right) ^{2} }$

$= $\sqrt{ \left (-5 \right ) ^{2}$ + 3 ^{2} } = $\sqrt{34}$ \approx 5.8$

Therefore, none of the integer answer choices are correct.

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Question

Find the distance between the two points and .

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Answer

The distance between two points can be found with the equation $\sqrt{(x_2-x_$1)^2$+(y_2-y_$1)^2$}$ . Substituting in values you get . This means that the answer is $\sqrt{37}$ .

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Question

Find the distance between the midpoints of line A with the points and and line. B with the points and .

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Answer

Use the midpoint formula:

Remember points are written in the following format:

Substitute for line A

$\frac{\left( -3+3 \right)}{2}$ , $\frac{\left( 4+0 \right)}{2}$

$\frac{\left( 0\right)}{2}$ , $\frac{\left( 4\right)}{2}$

The midpoint of line A is .

Substitute for line B.

$\frac{\left( -2+2 \right)}{2}$ , $\frac{\left( -2 +2 \right)}{2}$

$\frac{\left( 0 \right)}{2}$ , $\frac{\left( 0 \right)}{2}$

The midpoint of line B is .

Now we can find the distance between these two points using the distance formula:

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

Substitute the using the known values for lines A and B.

$d= $\sqrt{\left( 0-0 \right $)^{2}$$ + \left( 0-4 \right $)^{2}$}$

Simplify.

$d= $\sqrt{\left ( 0\right $)^{2}$$ + \left ( -4 \right $)^{2}$}$

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

$d= $\sqrt{16}$$

The distance between the two midpoints of lines A and B is .

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Question

Find the distance between the following points:

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Answer

Use the equation to calculated the distance between two points:

where

we can find the distance.

$=$\sqrt{25+36}$$

$=$\sqrt{61}$$

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Question

$\small (12,20)(2,4)$

Find the length of the line between the two points provided using the distance formula.

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Answer

It is definately possible to find the distance from point A to point B, given the coordinates.

We can do this by using the formula:

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

The points provided can be plugged into this formula as follows:

$\ \small \small \small d=\small $$\sqrt{(12-2)^{2}$$$+(20-4)^{2}$} \ \ = $$\sqrt{10^{2}$$$+16^{2}$}\ \ = $\sqrt{100+256}$ \ \= \sqrt {356}$ .

This is the length.

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Question

$\small (4,6)(10,2)$

Find the length of the line between the two points provided using the distance formula.

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Answer

It is definately possible to find the distance from point A to point B, given the coordinates:

$\small (4,6)(10,2)$ .

We can do this by using the formula:

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

The points provided can be plugged into this formula as follows:

$\ \small \small \small \small d=\small $$\sqrt{(10-4)^{2}$$$+(2-6)^{2}$} \ \= $$\sqrt{6^{2}$$$+(-4)^{2}$} \ \= $\sqrt{36+16}$ \ \= \sqrt {52}$ .

This is the length.

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Question

$\small (4,5)(6,7)$

Find the length of the line between the two points provided using the distance formula.

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Answer

It is definately possible to find the distance from point A to point B, given the coordinates:

$\small (4,5)(6,7)$ .

We can do this by using the formula:

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

The points provided can be plugged into this formula as follows:

$\ \small \small \small \small \small d=\small $$\sqrt{(6-4)^{2}$$$+(7-5)^{2}$} \ \= $$\sqrt{2^{2}$$$+2^{2}$} \ \= $\sqrt{4+4}$ \ \= \sqrt 8$ .

This is the length.

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Question

$\small (10,4)(5,8)$

Find the length of the line between the two points provided using the distance formula.

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Answer

It is definately possible to find the distance from point A to point B, given the coordinates:

$\small (10,4)(5,8)$ .

We can do this by using the formula:

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

The points provided can be plugged into this formula as follows:

$\ \small \small \small \small \small d=\small $$\sqrt{(10-5)^{2}$$$+(4-8)^{2}$} \ \= $$\sqrt{5^{2}$$$+(-4)^{2}$} \ \= $\sqrt{25+16}$ \ \= \sqrt {41}$ .

This is the length.

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Question

$\small (12,7)(14,8)$

Find the length of the line between the two points provided using the distance formula.

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Answer

It is definately possible to find the distance from point A to point B, given the coordinates:

$\small (12,7)(14,8)$ .

We can do this by using the formula:

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

The points provided can be plugged into this formula as follows:

$\d=\small $$\sqrt{(14-12)^{2}$$$+(8-7)^{2}$} \ \= $$\sqrt{2^{2}$$$+1^{2}$} \ \= $\sqrt{4+1}$ \ \= \sqrt 5$ .

This is the length.

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Question

$\small (10,4)(8,2)$

Find the length of the line between the two points provided using the distance formula.

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Answer

It is definately possible to find the distance from point A to point B, given the coordinates:

$\small (10,4)(8,2)$ .

We can do this by using the formula:

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

The points provided can be plugged into this formula as follows:

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

This is the length.

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Question

$\small (10,4)(6,2)$

Find the length of the line between the two points provided using the distance formula.

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Answer

It is definately possible to find the distance from point A to point B, given the coordinates:

$\small (10,4)(6,2)$ .

We can do this by using the formula:

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

The points provided can be plugged into this formula as follows:

$\ d=\small $$\sqrt{(10-6)^{2}$$$+(4-2)^{2}$} \ \= $$\sqrt{4^{2}$$$+2^{2}$} \ \= $\sqrt{16+4}$ \ \= \sqrt {20}$ .

This is the length.

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Question

$\small (14,9)(12,7)$

Find the length of the line between the two points provided using the distance formula.

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Answer

It is definately possible to find the distance from point A to point B, given the coordinates:

$\small (14,9)(12,7)$ .

We can do this by using the formula:

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

The points provided can be plugged into this formula as follows:

$\\small d=\small $$\sqrt{(14-12)^{2}$$$+(9-7)^{2}$} \ \= $$\sqrt{2^{2}$$$+2^{2}$} \ \= $\sqrt{4+4}$ \ \= \sqrt 8$ .

This is the length.

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Question

$\small (20,4)(16,7)$

Find the length of the line between the two points provided using the distance formula.

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Answer

It is definately possible to find the distance from point A to point B, given the coordinates:

$\small (20,4)(16,7)$ .

We can do this by using the formula:

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

The points provided can be plugged into this formula as follows:

$\ \small d=\small $$\sqrt{(20-16)^{2}$$$+(4-7)^{2}$} \ \= $$\sqrt{4^{2}$$$+3^{2}$} \ \= $\sqrt{16+9}$ \ \= $\sqrt{25}$\ \=5$ .

This is the length.

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Question

$\small (14,8)(16,7)$

Find the length of the line between the two points provided using the distance formula.

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Answer

It is definately possible to find the distance from point A to point B, given the coordinates:

$\small (14,8)(16,7)$ .

We can do this by using the formula:

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

The points provided can be plugged into this formula as follows:

$\ \small d=\small $$\sqrt{(16-14)^{2}$$$+(7-8)^{2}$}\ \ = $$\sqrt{2^{2}$$$+(-1)^{2}$} \ \= $\sqrt{4+1}$\ \ = \sqrt 5$ .

This is the length.

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Question

$\small (20,38)(16,28)$

Find the length of the line between the two points provided using the distance formula.

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Answer

It is definately possible to find the distance from point A to point B, given the coordinates:

$\small (20,38)(16,28)$ .

We can do this by using the formula:

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

The points provided can be plugged into this formula as follows:

$\ \small d=\small $$\sqrt{(20-16)^{2}$$$+(38-28)^{2}$} \ \= $$\sqrt{4^{2}$$$+(10)^{2}$} \ \= $\sqrt{16+100}$ \ \= \sqrt {116}$ .

This is the length.

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Question

$\small (10,4)(14,9)$

Find the length of the line between the two points provided using the distance formula.

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Answer

It is definately possible to find the distance from point A to point B, given the coordinate:

$\small (10,4)(14,9)$

We can do this by using the formula:

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

The points provided can be plugged into this formula as follows:

$\ \small d=\small $$\sqrt{(14-10)^{2}$$$+(9-4)^{2}$} \ \= $$\sqrt{4^{2}$$$+5^{2}$} \ \= $\sqrt{16+25}$\ \ = \sqrt {41}$ .

This is the length.

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Question

You have to walk from the grocery store north blocks then west blocks. How far will you be from your original position?

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Answer

If the student is a visual learner they could draw out a journey of 5 units north then 6 units west. To find the distance between the original and final points they could draw a line connecting the two.

This would ultimately create a right triangle. To find the final distance you could use the Pythagorean Theorem. The longest side length is opposite of the largest angle, which is degrees.

$$\sqrt{61}$=7.81$ blocks

But the answer must be in the format of distance in miles. So we convert blocks to miles.

$\frac{1block}{0.25 miles}$=\frac{7.81blocks}{x miles}$

$x \cdot 1 = 0.25 \cdot 7.81$

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