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Algebra 1 : Points and Distance Formula

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Example Questions

Example Question #1 : Points And Distance Formula

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

Possible Answers:

Correct answer:

Explanation:

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

$\frac{5}{2}$

$8\sqrt{2}$

$2\sqrt{53}$

Infinite

Correct answer:

$2\sqrt{53}$

Explanation:

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

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

The two points are (x_1,y_1) and (x_2,y_2) . 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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Example Question #2 : How To Find The Length Of A Line With Distance Formula

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

Possible Answers:

$\sqrt{20}$

$\sqrt{7}$

Correct answer:

Explanation:

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

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

Possible Answers:

None of the other answers are correct.

Correct answer:

None of the other answers are correct.

Explanation:

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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Example Question #5 : How To Find The Length Of A Line With Distance Formula

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

Possible Answers:

$\sqrt{37}$

$\sqrt{5}$

Correct answer:

$\sqrt{37}$

Explanation:

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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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 (-3,4) and (3,4) and line. B with the points (-2,-2) and (2,2) .

Possible Answers:

$\frac{3}{2}$

$\frac{1}{2}$

Correct answer:

Explanation:

Use the midpoint formula:

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

Remember points are written in the following format:

(x,y)

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 (0,4) .

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 (0,0) .

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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Example Question #7 : How To Find The Length Of A Line With Distance Formula

Find the distance between the following points:

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

Possible Answers:

$\sqrt{65}$

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

$\sqrt{29}$

$\sqrt{61}$

Correct answer:

$\sqrt{61}$

Explanation:

Use the equation to calculated the distance between two points:

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

where

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

we can find the distance.

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

$=\sqrt{25+36}$

$=\sqrt{61}$

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

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

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

Possible Answers:

$\small \sqrt {192}$

$\small \sqrt {456}$

$\small \sqrt {356}$

$\small \sqrt {35}$

Correct answer:

$\small \sqrt {356}$

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:

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

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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Example Question #7 : How To Find The Length Of A Line With Distance Formula

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

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

Possible Answers:

$\small \sqrt {203}$

$\small \sqrt {62}$

$\sqrt {52}$

$\small \sqrt {19}$

Correct answer:

$\sqrt {52}$

Explanation:

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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Example Question #4 : Points And Distance Formula

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

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

Possible Answers:

$\small \sqrt 8$

$\small 18$

$\small 8$

$\small \sqrt {18}$

Correct answer:

$\small \sqrt 8$

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.

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Algebra 1 : Points and Distance Formula

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #1 : Points And Distance Formula

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

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

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

Example Question #5 : How To Find The Length Of A Line With Distance Formula

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

Example Question #7 : How To Find The Length Of A Line With Distance Formula

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

Example Question #7 : How To Find The Length Of A Line With Distance Formula

Example Question #4 : Points And Distance Formula

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