How to find the length of a line with distance formula

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

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1

Find the length of the line connecting the following two points. Simplify your answer.

and

$5\sqrt{5}$

$4\sqrt{5}$

$\sqrt{123}$

$\sqrt{127}$

Explanation

To solve this problem we need to remember the distance formula for points on a coordinate plane:

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

In this case, and

$d=\sqrt{(12-2)^{2}+(6-1)^{2}}$

$d=\sqrt{10^{2}+5^{2}}$

$d=\sqrt{100+25}=\sqrt{125}=5\sqrt{5}$

2

Find the length of the line with the following endpoints:

$\sqrt{29}$

$2\sqrt{27}$

$\sqrt{26}$

Explanation

In order to find the distance between two points on a line, use the formula below:

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

Plug in the points (2,1) and (7,3) and solve:

$\sqrt{(7-2)^{2}+(3-1)^{2}}=\sqrt{(5)^{2}+(2)^{2}}=\sqrt{25+4}=\sqrt{29}$

This gives a final answer of $\sqrt{29}$

3

Find the length of the line connecting the following two points. Simplify your answer.

and

$\sqrt{41}$

$\sqrt{40}$

$\sqrt{42}$

$\sqrt{43}$

Explanation

To solve this problem we need to remember the distance formula for points on a coordinate plane:

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

In this case, and

$d=\sqrt{(10-5)^{2}+(2-6)^{2}}$

$d=\sqrt{5^{2}+(-4)^{2}}$

$d=\sqrt{25+16}=\sqrt{41}$

4

Find the distance of the line connecting the pair of points

and .

Explanation

By the distance formula

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

where and

we have

$\sqrt{(5-0)^2+(12-0)^2 }=\sqrt{(5)^2+(12)^2}= \sqrt{25+144} = \sqrt{169} = 13$

5

Find the length of the line with the following endpoints:

$2\sqrt{130}$

$\sqrt{523}$

$10\sqrt{52}$

$4\sqrt{254}$

Explanation

In order to find the distance between two points on a line, use the formula below:

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

Plug in the points (1,2) and (15,20) and solve:

$\sqrt{(15-1)^{2}+(20-2)^{2}}= \sqrt{(14)^{2}+(18)^{2}}= \sqrt{196+324}= \sqrt{520}= 2\sqrt{130}$

This gives a final answer of $2\sqrt{130}$

6

What is the length of a line with endpoints of and ?

Explanation

The distance formula is just a reworking of the Pythagorean theorem: $l=\sqrt{\Delta x^2+\Delta y^2}$

Expand that.

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

Plug in our given values.

$l=\sqrt{(7-2)^2+(9--4)^2}$

$l=\sqrt{(5)^2+(13)^2}$

$l=\sqrt{25+169}$

$l=\sqrt{194}$

7

Find the length of the line with the following endpoints:

$\sqrt{178}$

$4\sqrt{89}$

$2\sqrt{76}$

$\sqrt{177}$

Explanation

In order to find the distance between two points on a line, use the formula below:

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

Plug in the points (-1,-6) and (12,-3) and solve:

$\sqrt{(12+1)^{2}+(-3+6)^{2}}=\sqrt{(13)^{2}+(3)^{2}}=\sqrt{169+9}=\sqrt{178}$

This gives a final answer of $\sqrt{178}$

8

A line is connected by the points and . What is the distance of this line?

$\sqrt{10}$

$2\sqrt2$

$2\sqrt5$

$2\sqrt3$

Explanation

Write the distance formula.

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

Substitute the values of the points inside the equation.

$d= \sqrt{(4-1)^2+(2-3)^2}$

Simplify by order of operations.

$d= \sqrt{(3)^2+(-1)^2} = \sqrt{9+1} = \sqrt{10}$

The length of the line is $\sqrt{10}$ .

9

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

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

$\small \sqrt 8$

$\small 8$

$\small \sqrt {18}$

$\small 18$

Explanation

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.

10

What is the distance between the following points?

Explanation

What is the distance between the following points?

To find distance, use distance formula:

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

Note, if you cannot recall distance formula think of Pythagorean theorem. When using distance formula, you are simply finding the length of the hypotenuse of a right triangle.

Anyway, start plugging in our points and simplify to the answer. We'll call our first point 1 and our second point 2

$d=\sqrt{(4--19)^2+(-6--53)^2}=\sqrt{(23)^2+(47)^2}=\sqrt{2738}\approx52.3$

So our answer is 52.3

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