Calculating the length of a line with distance formula - GMAT Quantitative Reasoning

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Question

What is the distance between the points and ?

Answer

Let's plug our coordinates into the distance formula.

$\sqrt{(2-7)^{2}+(5-17)^{2}}= \sqrt{(-5)^{2}+(-12)^{2}} = \sqrt{25+144}= \sqrt{169} = 13$

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Question

What is the distance between the points and ?

Answer

We need to use the distance formula to calculate the distance between these two points.

$\sqrt{(1-5)^{2}+(4-2)^{2}} = \sqrt{(-4)^{2}+(2)^{2}} =\sqrt{20}=\sqrt{4}\sqrt{5}=2\sqrt{5}$

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Question

A line segement on the coordinate plane has endpoints and . Which of the following expressions is equal to the length of the segment?

Answer

Apply the distance formula, setting

$x_{ 1} = A + 4, x_{2} = A ,y_{ 1} = B, y_{2} = B+3$ :

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

$d = \sqrt{[A-(A+4)]^{2}+[(B+3)-B)]^{2}}$

$d = \sqrt{(-4)^{2}+3^{2}} = \sqrt{16+9} = \sqrt{25} = 5$

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Question

What is distance between and ?

Answer

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

$= \sqrt{(6-1)^2+(7-5)^2}$

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

$= \sqrt{(25+4)}$

$\sqrt{29}$

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Question

Consider segment $\overline{JK}$ which passes through the points $\left ( 4,5\right )$ and $\left ( 144,75\right )$ .

Find the length of segment $\overline{JK}$ .

Answer

This question requires careful application of distance formula, which is really a modified form of Pythagorean theorem.

$\small d=\sqrt{(x-x')^2+(y-y')^2}$

Plug in everthing and solve:

$\small \small d=\sqrt{(144-4)^2+(75-7)^2}=\sqrt{24500}\approx156.5$

So our answer is 156.6

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Question

What is the length of a line segment that starts at the point and ends at the point ?

Answer

Using the distance formula for the length of a line between two points, we can plug in the given values and determine the length of the line segment by calculating the distance between the two points:

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

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

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

$d=\sqrt{25}$

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