This construction shows how to create a line segment whose length is the difference between two given segments.
This is very similar to Sum of line segments, except that the second segment is drawn on the other side of the first, effectively subtracting the lengths.
The two types of construction (add , subtract) can be combined in any way you like. For example you could find the resulting length of a+b–d–f.
What if the second segment is longer than the first?
If b is longer than a, then you will find the end of the construction to the left of the start:
The distance PQ is still the difference between the two segment lengths.
In arithmetic, this would result in a negative length, which does not exist in Euclidean geometry.
Segments cannot have a negative length.
So the precise definition of what this construction does is find the absolute value of a–b
The two vertical bars mean "absolute value" which is always positive, regardless of the way a–b comes out.
This is why the construction is titled "Difference of two segments" – it is not quite the same as arithmetic subtraction.
Proof
The proof of this construction is trivial. This is the same drawing as the last step in the above animation.
| Argument | Reason | |
|---|---|---|
| 1 | The segment PQ is congruent to the given segment a | Copied using the procedure in Copying a line segment. See that page for proof. |
| 2 | The segment RQ is congruent to the given segment b | As in (1) |
| 4 | PR is the difference of given segments a,b,c | From (1), (2). All segments are colinear and adjacent. |
- Q.E.D
Printable step-by-step instructions
The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.
Try it yourself
Click here for a printable worksheet containing two line segment copying problems. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.Other constructions pages on this site
Lines
- Introduction to constructions
- Copy a line segment
- Sum of n line segments
- Difference of two line segments
- Perpendicular bisector of a line segment
- Perpendicular from a line at a point
- Perpendicular from a line through a point
- Perpendicular from endpoint of a ray
- Divide a segment into n equal parts
- Parallel line through a point (angle copy)
- Parallel line through a point (rhombus)
- Parallel line through a point (translation)
Angles
- Bisecting an angle
- Copy an angle
- Construct a 30° angle
- Construct a 45° angle
- Construct a 60° angle
- Construct a 90° angle (right angle)
- Sum of n angles
- Difference of two angles
- Supplementary angle
- Complementary angle
- Constructing 75° 105° 120° 135° 150° angles and more
Triangles
- Copy a triangle
- Isosceles triangle, given base and side
- Isosceles triangle, given base and altitude
- Isosceles triangle, given leg and apex angle
- Equilateral triangle
- 30-60-90 triangle, given the hypotenuse
- Triangle, given 3 sides (sss)
- Triangle, given one side and adjacent angles (asa)
- Triangle, given two angles and non-included side (aas)
- Triangle, given two sides and included angle (sas)
- Triangle medians
- Triangle midsegment
- Triangle altitude
- Triangle altitude (outside case)
Right triangles
- Right Triangle, given one leg and hypotenuse (HL)
- Right Triangle, given both legs (LL)
- Right Triangle, given hypotenuse and one angle (HA)
- Right Triangle, given one leg and one angle (LA)
Triangle Centers
Circles, Arcs and Ellipses
- Finding the center of a circle
- Circle given 3 points
- Tangent at a point on the circle
- Tangents through an external point
- Tangents to two circles (external)
- Tangents to two circles (internal)
- Incircle of a triangle
- Focus points of a given ellipse
- Circumcircle of a triangle
Polygons
- Square given one side
- Square inscribed in a circle
- Hexagon given one side
- Hexagon inscribed in a given circle
- Pentagon inscribed in a given circle
Non-Euclidean constructions
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