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

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Question

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

Answer

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

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

$d=\sqrt{(3-0)^2+(4-0)^2}$

$d=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$

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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?

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).

$d=\sqrt{(6-(-8))^2+(-1-3)^2}$

$d=\sqrt{14^2+(-4)^2}$

$d=\sqrt{196+14}$

$d=\sqrt{212}$

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

$d=2\sqrt{53}$

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 ?

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 .

Answer

Use the distance formula, with $x_{1} =2,x_{2} =7,y_{1} =9,y_{2} =6$ :

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

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

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

Therefore, none of the integer answer choices are correct.

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Question

Find the distance between the two points and .

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 $\sqrt{(2+4)^2+(1-2)^2}$ . 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 .

Answer

Use the midpoint formula:

$\frac{x_{2}+x_{1}}{2}, \frac{y_{2}+y_{1}}{2}$

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:

Answer

Use the equation to calculated the distance between two points:

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

where

we can find the distance.

$d=\sqrt{(-2-3)^{2}+(7-1)^{2}}$

$=\sqrt{25+36}$

$=\sqrt{61}$

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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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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?

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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