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

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Question

A line segment has an endpoint at ; what is its length?

Its other endpoint is

Its midpoint is

Answer

Given the other endpoint, you can use the distance formula to find the length of the segment:

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

$d = \sqrt{\left (7-1 \right) ^{2}+\left (-9-(-1) \right) ^{2}}$

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

Given the midpoint, you can use the distance formula to find the distance from the first endpoint to the midpoint, then double that to get the length of the segment:

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

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

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

The total length is twice that, or 10.

The answer is that either statement alone is sufficient to answer the question.

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Question

Note: Figure NOT drawn to scale.

Give .

Statement 1:

Statement 2:

Answer

If you know only that , then you know that and , but you still need , or a way finding it.

If you know only that , you still know only that , but you don't know their actual lengths.

If you know both facts, then you know

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Question

Consider segment .

I) Point can be found at the point .

II) Segment had a length of units.

Find the coordinates of point .

Answer

Statement I gives us a point.

Statement II gives us the length of the segment.

We are asked to find the coordinates of the other end of the segment. However, we will need more information. Even with all of our information, we have no clue as to the orientation of the line. It could be 14 units straight up and down, it could be a perfectly horizontal line, or something inbetween, thus our answer is:

Neither I nor II are sufficient to answer the question. More information is needed.

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Question

Find the length of Segment YZ

I) Point Y is located at the point .

II) Point Z has a y-coordinate twice that of Point Y and an x-coordinate one-third of Point Y.

Answer

To find the length of a segment, use the distance formula. The distance formula is given by the following:

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

Where your 's and 's correspond to the coordinates of the endpoints.

To find the length of Segment YZ, we need the endpoints.

Statement I gives you Point Y's coordinates.

Statement II relates Point Z's coordinates to Point Y's coordinates. Thus, we can find the point Z using Statement II.

Therefore, we need both.

Recap:

Find the length of Segment YZ

I) Point Y is located at the point .

II) Point Z has a y-coordinate twice that of Point Y, and an x-coordinate one-third of Point Y.

Use Statement II along with Statement I to find the coordinates of Point Z:

Then, use distance formula to find the length of Segment YZ:

$d=\sqrt{(3-9)^2+(14-7)^2}=\sqrt{36+49}=\sqrt{85}$

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Question

Consider segment

I) Endpoint is located at the point .

Ii) Endpoint has an x-coordinate twice that of and a y-coordinate 15 times that of H.

What is the length of ?

Answer

To find the length of a segment, we need both endpoints.

Statement I gives us one endpoint.

Statement II relates and , allowing us to find the second endpoint.

Thus, we need both. Once we have both endpoints, distance is easily calculated via the distance formula or the Pythagorean theorem.

Using Statement II, we find the second endpoint to be . Use the distance formula to find your answer:

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

$d=\sqrt{(6-12)^2+(14-210)^2}=\sqrt{36+38416}=\sqrt{38452}$

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