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Algebra 1 : Points and Distance Formula

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Example Questions

Example Question #3899 : Algebra 1

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

(12,2) and (10,1)

Possible Answers:

$\sqrt{5}$

$\sqrt{3}$

$\sqrt{6}$

$2\sqrt{2}$

Correct answer:

$\sqrt{5}$

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, (x_1,y_1)=(12,2) and (x_2,y_2)=(10,1)

$d=\sqrt{(12-10)^{2}+(2-1)^{2}}$

$d=\sqrt{2^{2}+1^{2}}$

$d=\sqrt{4+1}=\sqrt{5}$

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Example Question #3900 : Algebra 1

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

(-15,-5) and (-4,-1)

Possible Answers:

$\sqrt{137}$

$\sqrt{134}$

$\sqrt{136}$

$\sqrt{135}$

Correct answer:

$\sqrt{137}$

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, (x_1,y_1)=(-15,-5) and (x_2,y_2)=(-4,-1)

$d=\sqrt{((-4)-(-15))^{2}+((-1)-(-5))^{2}}$

$d=\sqrt{11^{2}+4^{2}}$

$d=\sqrt{121+16}=\sqrt{137}$

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Example Question #31 : How To Find The Length Of A Line With Distance Formula

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

(-12,16) and (-7,4)

Possible Answers:

$\sqrt{173}$

$\sqrt{171}$

Correct answer:

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, (x_1,y_1)=(-7,4) and (x_2,y_2)=(-12,16)

$d=\sqrt{((-7)-(-12))^{2}+(4-16)^{2}}$

$d=\sqrt{5^{2}+(-12)^{2}}$

$d=\sqrt{25+144}=\sqrt{169}=13$

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Example Question #32 : How To Find The Length Of A Line With Distance Formula

A line passes through the points (7,25) and (12,0) . What is the distance between these two points?

Possible Answers:

23.7

25.5

26.8

Correct answer:

25.5

Explanation:

The question is merely asking the distance between two points. This kind of problem can be quickly solved for by using the distance formula:

$d= \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$ , where is distance and x_1 , x_2 , y_1 , and y_2 come from the given points.

This problem merely needs to have the and values substituted in for so we can solve for .

Arbitrarily assigning (x_1,y_1) and (x_2,y_2) , we substitute in our values as follows:

$d=\sqrt{(12-7)^2+(0-25)^2}$

$d=\sqrt{(5)^2+(-25)^2}$

$d=\sqrt{25+625}$

$d=\sqrt{650}$

$d=25.495 \approx {\color{Blue} 25.5}$

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Example Question #33 : How To Find The Length Of A Line With Distance Formula

A line passes through (1,3) and (-9,32) . What's the distance between these two points?

Possible Answers:

33.6

30.6

31.4

27.9

30.7

Correct answer:

30.7

Explanation:

The question is merely asking the distance between two points. This kind of problem can be quickly solved for by using the distance formula:

$d= \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$ , where is distance and x_1 , x_2 , y_1 , and y_2 come from the given points.

This problem merely needs to have the and values substituted in for so we can solve for .

Arbitrarily assigning (x_1,y_1) and (x_2,y_2) , we substitute in our values as follows:

$d=\sqrt{(1+9)^2+(3-32)^2}$

$d=\sqrt{(10)^2+(-29)^2}$

$d=\sqrt{100+841}$

$d=\sqrt{941}$

$d=30.6757 \approx {\color{Blue} 30.7}$

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Example Question #612 : Functions And Lines

What is the distance between the following points?

Possible Answers:

52.3

23.5

17.4 36.1

33.6

Correct answer:

52.3

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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Example Question #35 : How To Find The Length Of A Line With Distance Formula

What is the length of the distance between the points and ?

Possible Answers:

12.5

16.25

Correct answer:

Explanation:

What is the length of the distance between the points and ?

Find distance with distance formula, which is quite similar to Pythagorean Theorem

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

Now, let's call point 1 and point 2, then let's plug in and find d!

$d=\sqrt{(9-18)^2+(15-3)^2}=\sqrt{81+144}=\sqrt{225}=15$

So the distance is 15

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Example Question #36 : How To Find The Length Of A Line With Distance Formula

Find the distance between the following points:

(14,25)(38,7)

Possible Answers:

Correct answer:

Explanation:

Find the distance between the following points:

(14,25)(38,7)

To find the distance between two points, use distance formula.

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

Distance formula is closely related to Pythagorean theorem. Pythagorean Theorem is:

a^2+b^2=c^2

Which can be rewritten as:

$c=\sqrt{a^2+b^2}$

Now, in distance formula, we are essentially finding the hypotenuse of a right triangle.

Anyway, to find the distance, we simply need to plug in the points we are given and simplify:

$d=\sqrt{(38-14)^2+(7-25)^2}=\sqrt{(24^2)+(-18)^2}$

Continue

$\sqrt{(24^2)+(-18)^2}=\sqrt{576+324}=\sqrt{900}=30$

So our hypotenuse (distance) is 30.

If you are really observant, you can see that the other two sides of the triangles are 24 and 18.

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Example Question #37 : How To Find The Length Of A Line With Distance Formula

A line is connected by the points (1,3) and (4,2) . What is the distance of this line?

Possible Answers:

$2\sqrt3$

$\sqrt{10}$

$2\sqrt5$

$2\sqrt2$

Correct answer:

$\sqrt{10}$

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 $\sqrt{10}$ .

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Example Question #31 : How To Find The Length Of A Line With Distance Formula

Find the distance between these two points using the distance formula:

(2, 6) and (-4, 8)

Possible Answers:

$\sqrt{40}$

$\sqrt{6}$

$\sqrt{10}$

Correct answer:

$\sqrt{40}$

Explanation:

To find the distance between two points using the distance formula, we use the following formula:

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

where (x_1, y_1) and (x_2, y_2) are the given points. So, we can substitute the points (2, 6) and (-4, 8) .

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

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

$d = \sqrt{36 + 4}$

$d = \sqrt{40}$

Therefore, the distance is $\sqrt{40}$ .

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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 #3899 : Algebra 1

Example Question #3900 : Algebra 1

Example Question #31 : How To Find The Length Of A Line With Distance Formula

Example Question #32 : How To Find The Length Of A Line With Distance Formula

Example Question #33 : How To Find The Length Of A Line With Distance Formula

Example Question #612 : Functions And Lines

Example Question #35 : How To Find The Length Of A Line With Distance Formula

Example Question #36 : How To Find The Length Of A Line With Distance Formula

Example Question #37 : How To Find The Length Of A Line With Distance Formula

Example Question #31 : How To Find The Length Of A Line With Distance Formula

All Algebra 1 Resources

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