Algebra 1 : How to find the length of a line with distance formula

Study concepts, example questions & explanations for Algebra 1

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Example Questions

Example Question #1 : How To Find The Length Of A Line With Distance Formula

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

Possible Answers:

25

7

1

49

5

Correct answer:

5

Explanation:

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

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

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

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

Example Question #1 : Points And Distance Formula

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?

Possible Answers:

\displaystyle 2\sqrt{53}

\displaystyle 8\sqrt{2}

Infinite

\displaystyle 3

\displaystyle \frac{5}{2}

Correct answer:

\displaystyle 2\sqrt{53}

Explanation:

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

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

The two points are \displaystyle (x_1,y_1) and \displaystyle (x_2,y_2).  In our case, the points are (–8, 3) and (6, –1).

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

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

\displaystyle d=\sqrt{196+14}

\displaystyle d=\sqrt{212}

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

\displaystyle d=2\sqrt{53}

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

Example Question #2 : Points And Distance Formula

What is the distance between the points \displaystyle (7,8) and \displaystyle (10,4)?

Possible Answers:

\displaystyle 3

\displaystyle \sqrt{20}

\displaystyle \sqrt{7}

\displaystyle 5

\displaystyle 6

Correct answer:

\displaystyle 5

Explanation:

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

Example Question #1 : How To Find The Length Of A Line With Distance Formula

Find the length of the line segment with endpoints at \displaystyle (2,9) and \displaystyle (7,6)

Possible Answers:

\displaystyle 16

\displaystyle 8

\displaystyle 4

\displaystyle 17

None of the other answers are correct.

Correct answer:

None of the other answers are correct.

Explanation:

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

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

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

Therefore, none of the integer answer choices are correct.

Example Question #1 : How To Find The Length Of A Line With Distance Formula

Find the distance between the two points \displaystyle (-4,2) and \displaystyle (2,1).

Possible Answers:

\displaystyle \sqrt{37}

\displaystyle 7

\displaystyle 2

\displaystyle 49

\displaystyle \sqrt{5}

Correct answer:

\displaystyle \sqrt{37}

Explanation:

The distance between two points can be found with the equation \displaystyle \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. Substituting in values you get \displaystyle \sqrt{(2+4)^2+(1-2)^2}. This means that the answer is \displaystyle \sqrt{37}.

Example Question #6 : How To Find The Length Of A Line With Distance Formula

Find the distance between the midpoints of line A with the points \displaystyle (-3,4) and \displaystyle (3,4) and line. B with the points \displaystyle (-2,-2) and \displaystyle (2,2).

Possible Answers:

\displaystyle \frac{3}{2}

\displaystyle \frac{1}{2}

\displaystyle 3

\displaystyle 4

\displaystyle 2

Correct answer:

\displaystyle 4

Explanation:

Use the midpoint formula:

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

Remember points are written in the following format:

\displaystyle (x,y)

Substitute for line A

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

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

The midpoint of line A is \displaystyle (0,4).

Substitute for line B.

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

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

The midpoint of line B is \displaystyle (0,0).

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

\displaystyle 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.

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

Simplify.

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

\displaystyle d= \sqrt{16}

The distance between the two midpoints of lines A and B is \displaystyle 4.

Example Question #2 : How To Find The Length Of A Line With Distance Formula

Find the distance between the following points: 

\displaystyle (3,1), \: (-2,7)

Possible Answers:

\displaystyle \left(\frac{1}{2},4\right)

\displaystyle \sqrt{61}

\displaystyle 5

\displaystyle \sqrt{29}

\displaystyle \sqrt{65}

Correct answer:

\displaystyle \sqrt{61}

Explanation:

Use the equation to calculated the distance between two points: 

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

where

\displaystyle (3,1)=(x_1,y_1), \: (-2,7)=(x_2,y_2)

we can find the distance.

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

   \displaystyle =\sqrt{25+36}

   \displaystyle =\sqrt{61}

Example Question #1 : How To Find The Length Of A Line With Distance Formula

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

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

Possible Answers:

\displaystyle \small \sqrt {192}

\displaystyle \small \sqrt {356}

\displaystyle \small \sqrt {35}

\displaystyle \small \sqrt {456}

Correct answer:

\displaystyle \small \sqrt {356}

Explanation:

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: 

\displaystyle \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:

\displaystyle \\ \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.  

Example Question #2 : How To Find The Length Of A Line With Distance Formula

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

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

Possible Answers:

\displaystyle \small \sqrt {19}

\displaystyle \sqrt {52}

\displaystyle \small \sqrt {203}

\displaystyle \small \sqrt {62}

Correct answer:

\displaystyle \sqrt {52}

Explanation:

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

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

We can do this by using the formula: 

\displaystyle \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: 

\displaystyle \\ \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. 

Example Question #2 : How To Find The Length Of A Line With Distance Formula

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

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

Possible Answers:

\displaystyle \small \sqrt {18}

\displaystyle \small \sqrt 8

\displaystyle \small 8

\displaystyle \small 18

Correct answer:

\displaystyle \small \sqrt 8

Explanation:

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

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

We can do this by using the formula:

\displaystyle \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: 

\displaystyle \\ \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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