|
Triangle given three sides (SSS)
Geometry construction using a compass and straightedge
Multiple triangles possible
It is possible to draw more than one triangle that has three sides with the given lengths.
For example in the figure below, given the base AB, you can draw four triangles that meet the requirements.
All four are correct in that they satisfy the requirements, and are
congruent to each other.
Note: This construction is not always possible
See figure on the right.
If two sides add to less than the third, no triangle is possible.
| After doing this |
Your work should look like this |
| Start with three line segments that will be the three sides of the triangle ABC. |
 |
| 1. Mark a point A that will be one vertex of the new triangle. |
 |
| 2. Set the compass width to the length of the segment AB.
This will become the base of the new triangle. |
 |
| 3. With the compass point on A, make an arc near the future vertex B of the triangle. |
 |
| 4. Mark a point B on this arc. This will become the next vertex of the new triangle. |
 |
| 5. Set the compass width to the length of the line segment AC. |
 |
| 6. Place the compass point on A and make an arc in the vicinity of where
the third vertex of the triangle (C) will be. All points along this arc are the distance AC from A,
but we do not yet quite know exactly where the vertex C is. |
 |
| 7. Use the compass to measure the length of the segment BC,
the length of the third side of the triangle. |
 |
| 8. From point B, draw an arc crossing the first.
Where these intersect is the vertex C of the triangle |
 |
| 9. Finally, draw the three sides AB, AC, and BC of the new triangle. |
 |
| 10. Done. The blue triangle ABC has each side
congruent to the the corresponding line segment. |
 |
Proof
The image below is the final drawing above with the red items added.
| |
Argument |
Reason |
| 1 |
Line segment LM is congruent to AB. |
Drawn with the same compass width. See Copying a line segment |
| 2 |
The third vertex N of the triangle must lie somewhere on arc P. |
All points on arc P are distance AC from L since the arc was drawn with the compass width set to AC. |
| 3 |
The third vertex N of the triangle must lie somewhere on arc Q. |
All points on arc Q are distance BC from M since the arc was drawn with the compass width set to BC. |
| 4 |
The third vertex N must lie where the two arcs intersect |
Only point that satisfies 2 and 3. |
| 5 |
Triangle LMN satisfies the three side lengths given.
LM is congruent to AB,
LN is congruent to AC,
MN is congruent to BC,
| |
- Q.E.D
Try it yourself
Click here for a printable worksheet containing two triangle construction problems where you are given the three side lengths.
When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.
Constructions pages on this site
Lines
Angles
Triangles
Triangle Centers
Circles, Arcs and Ellipses
Polygons
Non-Euclidean constructions
(C) 2009 Copyright John Page
|
|
