Algebra 1 : Equations of Lines

Study concepts, example questions & explanations for Algebra 1

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Example Questions

Example Question #53 : How To Find The Length Of A Line With Distance Formula

Find the length of the line with the following endpoints:

\(\displaystyle (6,7)(-6,-7)\)

Possible Answers:

\(\displaystyle 2\sqrt{85}\)

\(\displaystyle 2\sqrt{87}\)

\(\displaystyle \sqrt{341}\)

\(\displaystyle \sqrt{337}\)

Correct answer:

\(\displaystyle 2\sqrt{85}\)

Explanation:

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

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

Plug in the points (6,7) and (-6,-7) and solve:

\(\displaystyle \sqrt{(6+6)^{2}+(7+7)^{2}}=\sqrt{(12)^{2}+(14)^{2}}=\sqrt{144+196}=\sqrt{340}=2\sqrt{85}\)

 

This gives a final answer of \(\displaystyle 2\sqrt{85}\)

Example Question #54 : How To Find The Length Of A Line With Distance Formula

What is the distance of the line connected by the points \(\displaystyle (2,8)\) and \(\displaystyle (-3,6)\)?

Possible Answers:

\(\displaystyle 2\sqrt{5}\)

\(\displaystyle \sqrt{31}\)

\(\displaystyle \sqrt{29}\)

\(\displaystyle 3\sqrt{5}\)

\(\displaystyle \sqrt{21}\)

Correct answer:

\(\displaystyle \sqrt{29}\)

Explanation:

Use the distance formula to determine the line connected by the two points.

The distance formula is:

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

Let:

\(\displaystyle (x_1,y_1) = (2,8)\)

\(\displaystyle (x_2,y_2) =(-3,6)\)

Substitute the points into the formula.

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

Simplify the inner terms.

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

\(\displaystyle d= \sqrt{25+4} = \sqrt{29}\)

The distance of the line connect by the two points is:  \(\displaystyle \sqrt{29}\)

Example Question #55 : How To Find The Length Of A Line With Distance Formula

Find the length of the line with the following endpoints:

\(\displaystyle (-5,-3)(4,2)\)

Possible Answers:

\(\displaystyle 4\sqrt{29}\)

\(\displaystyle 4\sqrt{27}\)

\(\displaystyle 4\sqrt{28}\)

\(\displaystyle \sqrt{106}\)

Correct answer:

\(\displaystyle \sqrt{106}\)

Explanation:

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

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

Plug in the points (-5,-3) and (4,2) and solve:

\(\displaystyle \sqrt{(4+5)^{2}+(2+3)^{2}}=\sqrt{(9)^{2}+(5)^{2}}=\sqrt{81+25}=\sqrt{106}\)

 

This gives a final answer of \(\displaystyle \sqrt{106}\)

Example Question #56 : How To Find The Length Of A Line With Distance Formula

Find the length of the line with the following endpoints:

\(\displaystyle (4,9)(-2,-1)\)

Possible Answers:

\(\displaystyle 9\sqrt{5}\)

\(\displaystyle 2\sqrt{73}\)

\(\displaystyle \sqrt{136}\)

\(\displaystyle \sqrt{147}\)

Correct answer:

\(\displaystyle \sqrt{136}\)

Explanation:

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

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

Plug in the points (4,9) and (-2,-1) and solve:

\(\displaystyle \sqrt{(9+1)^{2}+(4+2)^{2}}=\sqrt{(10)^{2}+(6)^{2}}=\sqrt{100+36}=\sqrt{136}\)

 

This gives a final answer of \(\displaystyle \sqrt{136}\)

Example Question #57 : How To Find The Length Of A Line With Distance Formula

Find the length of the line between the two points

\(\displaystyle (1, 5)\) and \(\displaystyle (-3, 4)\)

Possible Answers:

\(\displaystyle \sqrt{12}\)

\(\displaystyle \sqrt{17}\)

\(\displaystyle \sqrt{3}\)

\(\displaystyle \sqrt{85}\)

\(\displaystyle \sqrt{5}\)

Correct answer:

\(\displaystyle \sqrt{17}\)

Explanation:

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

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

where \(\displaystyle (x_1, y_1)\) and \(\displaystyle (x_2, y_2)\) are the points given.

 

Using the given points

\(\displaystyle (1, 5)\) and \(\displaystyle (-3, 4)\)

we can substitute into the formula.  We get

 

\(\displaystyle d = \sqrt{(-3-1)^2 + (4-5)^2}\)

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

\(\displaystyle d = \sqrt{16 + 1}\)

\(\displaystyle d = \sqrt{17}\)

 

Therefore, the distance between the points \(\displaystyle (1, 5)\) and \(\displaystyle (-3, 4)\) is \(\displaystyle \sqrt{17}\).

Example Question #58 : How To Find The Length Of A Line With Distance Formula

Find the length of the line between the points

\(\displaystyle (-3, -3)\) and \(\displaystyle (-5, -8)\)

Possible Answers:

\(\displaystyle \sqrt{7}\)

\(\displaystyle \sqrt{185}\)

\(\displaystyle \sqrt{-7}\)

\(\displaystyle \sqrt{37}\)

\(\displaystyle \sqrt{29}\)

Correct answer:

\(\displaystyle \sqrt{29}\)

Explanation:

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

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

where \(\displaystyle (x_1, y_1)\) and \(\displaystyle (x_2, y_2)\) are the points given.

 

Using the given points

\(\displaystyle (-3, -3)\) and \(\displaystyle (-5, -8)\)

we can substitute into the formula.  We get

 

\(\displaystyle d = \sqrt{(-5 - -3)^2 + (-8 - -3)^2}\)

\(\displaystyle d = \sqrt{(-5+3)^2 + (-8+3)^2}\)

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

\(\displaystyle d = \sqrt{4 + 25}\)

\(\displaystyle d = \sqrt{29}\)

 

Therefore, the distance between the points \(\displaystyle (-3, -3)\) and \(\displaystyle (-5, -8)\) is \(\displaystyle \sqrt{29}\).

Example Question #59 : How To Find The Length Of A Line With Distance Formula

If the line \(\displaystyle \overline{AB}\) is connected by the points \(\displaystyle (8,3)\) and \(\displaystyle (-3,5)\), what is the exact length of this line?

Possible Answers:

\(\displaystyle \sqrt{89}\)

\(\displaystyle 5\sqrt5\)

\(\displaystyle \sqrt{39}\)

\(\displaystyle \sqrt{29}\)

\(\displaystyle 5\sqrt2\)

Correct answer:

\(\displaystyle 5\sqrt5\)

Explanation:

Use the distance formula to determine the distance of this line.

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

Substitute the points into the equation.

\(\displaystyle d=\sqrt{(5-3)^2+(-3-8))^2}\)

Simplify the radical.

\(\displaystyle d=\sqrt{(2)^2+(-11)^2} = \sqrt{4+121} = \sqrt{125}\)

Pull out the common factors.  This radical can still be reduced.

\(\displaystyle \sqrt{125} = \sqrt{25}\cdot \sqrt5 =5\sqrt5\)

The exact length of the line is:  \(\displaystyle 5\sqrt5\)

Example Question #60 : How To Find The Length Of A Line With Distance Formula

Steven and Joel are on a massive grid. Steven is located at point \(\displaystyle (1,2)\) on the grid and Joel is located at \(\displaystyle (-1,-2)\). How far away from each other are they?

Possible Answers:

\(\displaystyle 5\sqrt{2}units\)

\(\displaystyle 2\sqrt{5}units\)

\(\displaystyle 20units\)

None of these

\(\displaystyle 3units\)

Correct answer:

\(\displaystyle 2\sqrt{5}units\)

Explanation:

The distance formula is:

\(\displaystyle d= \sqrt{(x_{1}-x_{_{2}})^2+(y_{1}-y_2)^2}\)

Plug in these points into the distance formula.

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

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

\(\displaystyle d=\sqrt{20}\)

Reduce:

\(\displaystyle d=\sqrt{4}*\sqrt{5}\)

\(\displaystyle d=2\sqrt{5}\)

Example Question #61 : How To Find The Length Of A Line With Distance Formula

Steven and Joel are on a massive grid. Steven is located at point \(\displaystyle (1,2)\) on the grid and Joel is located at \(\displaystyle (-2,-2)\). How far away from each other are they?

Possible Answers:

\(\displaystyle 5units\)

Cannot be determined

\(\displaystyle 2\sqrt5units\)

\(\displaystyle 20units\)

\(\displaystyle 5\sqrt{2}units\)

Correct answer:

\(\displaystyle 5units\)

Explanation:

The distance formula is:

\(\displaystyle d= \sqrt{(x_{1}-x_{_{2}})^2+(y_{1}-y_2)^2}\)

Plug in Seven and Joel's coordinates into this formula.

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

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

\(\displaystyle d=\sqrt{25}=5units\)

Example Question #62 : How To Find The Length Of A Line With Distance Formula

What is the distance of the line connected by the points \(\displaystyle (-3,-6)\) and \(\displaystyle (7,-3)\)?

Possible Answers:

\(\displaystyle \sqrt{97}\)

\(\displaystyle \sqrt{109}\)

\(\displaystyle 5\)

\(\displaystyle \sqrt{7}\)

\(\displaystyle \sqrt{17}\)

Correct answer:

\(\displaystyle \sqrt{109}\)

Explanation:

Write the distance formula.

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

Substitute the given points into the formula.

\(\displaystyle D=\sqrt{(7-(-3))^2+(-3-(-6))^2}\)

Simplify the terms inside the parentheses.

\(\displaystyle D=\sqrt{(10)^2+(3)^2} =\sqrt{100+9} = \sqrt{109}\)

The answer is: \(\displaystyle \sqrt{109}\)

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