An equilateral triangle is a triangle with three equal side lengths and three equal angles. A compass and straightedge construction creates one exactly without measuring angles or using a ruler scale. This matters because it shows how geometric tools can copy lengths and prove shapes from definitions.
The two-arc method is one of the clearest examples of precise construction from simple rules.
Start with a base segment AB, then set the compass width equal to AB. Draw one arc centered at A and another arc centered at B, using the same compass width. Where the arcs intersect, label the point C and connect C to A and C to B.
Since C is the same distance from A as AB and the same distance from B as AB, AC = AB and BC = AB, so all three sides are equal.
Key Facts
- An equilateral triangle has AB = BC = CA.
- Each interior angle of an equilateral triangle measures 60 degrees.
- A compass copies a distance by keeping the same radius.
- Circle centered at A with radius AB contains all points P such that AP = AB.
- Circle centered at B with radius AB contains all points P such that BP = AB.
- If C is an intersection of the two arcs, then AC = AB and BC = AB, so AC = BC = AB.
Vocabulary
- Equilateral triangle
- A triangle whose three side lengths are all equal.
- Compass
- A drawing tool used to make circles or arcs with a fixed radius.
- Straightedge
- A tool used to draw straight lines without using measurement marks.
- Arc
- A connected part of a circle drawn with a compass.
- Radius
- The distance from the center of a circle to any point on the circle.
Common Mistakes to Avoid
- Changing the compass width between arcs is wrong because both arcs must use the same radius AB to copy the base length exactly.
- Using a marked ruler to measure the third point is wrong because a straightedge construction should rely on drawing lines, not measuring lengths.
- Connecting the wrong arc intersection point is wrong if the chosen point is not an intersection of both arcs, because then it may not be the same distance from A and B.
- Assuming the triangle is equilateral just because it looks symmetric is wrong because the proof depends on equal radii, not appearance.
Practice Questions
- 1 Segment AB is 6 cm long. Using the two-arc construction, what are the lengths of AC and BC in the completed equilateral triangle?
- 2 An equilateral triangle is constructed on base AB = 9.5 cm. What is the perimeter of the triangle?
- 3 Explain why the intersection point C of two arcs with centers A and B and radius AB guarantees that triangle ABC is equilateral.