A line is a collection of points that extends without limit in a straight formation. A line can be named by a single letter, like line l, or it can be named according to two points that it contains, like line AB. The second way of naming a line indicates an important property common to all lines: any two points in space determine a line. For example, given two points, J and K:
a line is determined:
This line is called JK.
Line Segments
A line segment is a section of a line. It is named and determined by its endpoints. Unlike a line, whose length is infinite, a line segment has finite length. Line segment AB is pictured below.
Distance and Midpoint of a Line Segment
The midpoint of a line segment is the point on the segment that is equidistant (the same distance) from each endpoint. Because a midpoint splits a line segment into two equal halves, the midpoint is said to bisect the line segment.
Because a midpoint cuts a line segment in half, knowing the distance between the midpoint and one endpoint of a line segment allows you to calculate the length of the entire line segment. For example, if the distance from one endpoint to the midpoint of a line segment is 5, the length of the whole line segment is 10.
The Math IC test often asks questions that focus on this property of midpoints. The Math IC writers usually make their questions a little trickier though, by including multiple midpoints. Take a look:
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X is the midpoint of WZ and Y is the midpoint of XZ. If M is the midpoint of XY and MY = 3, what is the length of WX? | |
All the midpoints flying around in this question can get quite confusing. Instead of trying to visualize what is being described in your head, draw a sketch of what the question describes.
Once you’ve drawn a sketch, you can see how the three midpoints, and the new line segments that the midpoints create, all relate to each other.
- Since X is the midpoint of WZ, you know that WX = XZ and that both WX and XY are equal to 1⁄2WZ.
- Since Y is the midpoint of XZ, you know that XY = YZ and that both XY and YZ are equal to 1⁄2XZ and 1⁄4WZ.
- Since M is the midpoint of XY, you know that XM = MY and that both XM and MY are equal to 1⁄2XY and 1⁄8WZ.
Please note that you don’t have to write out these relationships when answering this sort of question. If you draw a good sketch, it’s possible to see the relationships.
Once you know the relationships, you can solve the problem. For this question, you know that MY is equal to 1⁄8WZ. Since, as the question tells you, MY = 3, you can calculate that WZ = 24. The question asks for the length of WX, which is equal to 1⁄2WZ, so WX = 12.
Angles
