A Tale of Three Intersecting Lines · Lesson 1 of 10
Equilateral Triangles
“See how two compass arcs locate the third vertex of a triangle with three equal sides.”
• Identify a triangle’s vertices, sides, and interior angles. • Name a triangle and its angles using the vertex labels. • Explain why three points on one straight line do not form a triangle. • Construct an equilateral triangle using a ruler and compass. • Explain why the compass construction produces three equal sides.
Understanding a Triangle
A triangle is a closed shape made by joining three points with three straight line segments. The points must not all lie on one straight line. Think of three pegs joined by a tight string: if the pegs form three corners, the string encloses a region. If all three pegs lie along one straight path, there is no triangular region.
A corner point of a shape. A triangle has three vertices; “vertices” is the plural of “vertex”.
In triangle ABC, the vertices are A, B, and C. Its sides are AB, BC, and CA. The symbol Δ means “triangle”, so ΔABC is read as “triangle ABC”. You can name this same triangle ΔBCA or ΔCAB, because each name still uses the same three vertices.
Each pair of sides meeting at a vertex makes an interior angle. The angle at A can be written ∠CAB or ∠BAC: the middle letter names the vertex. If there is no confusion about which angle is meant, we write simply ∠A. Turning the drawing or using different letters does not change these ideas.
Equilateral Triangles
Some triangles have sides of different lengths. An equilateral triangle has all three sides equal, so we must place its third vertex at exactly the correct distance from both ends of the base. This makes it a useful first construction: the distances are easy to state, but their simultaneous satisfaction needs a careful method.
A triangle in which all three sides have the same length.
Suppose AB is 4 cm long. If you use a ruler to place C so that AC is 4 cm, BC may still be too short or too long. One correct distance does not guarantee the other. Moving C repeatedly until both distances seem right is possible, but it is slow and difficult to make accurate.
Constructing with a Compass
A compass keeps one fixed distance while its pencil moves. Every point on an arc drawn from A with a compass opening of 4 cm is 4 cm from A. Instead of guessing one position for C, we first draw all the nearby positions that satisfy one distance, then use a second arc to satisfy the other distance.
- Draw AB = 4 cm with a marked ruler.
- Open the compass to 4 cm. With its point at A, draw an arc above AB.
- Keep the opening at 4 cm. With its point at B, draw a second arc that crosses the first. Name an intersection C.
- Join A to C and B to C using the ruler.
Why does this work? AB was drawn as 4 cm. C lies on A’s arc, so AC = 4 cm. C also lies on B’s arc, so BC = 4 cm. All three sides are equal, which is exactly the definition of an equilateral triangle. The compass establishes the distances; the ruler joins the required points.
If you draw complete circles instead of upper arcs, there is also an intersection below AB. Either intersection gives an equilateral triangle. They lie on opposite sides of the same base. Short arcs are sufficient as long as they are long enough to cross.
Do not change the compass opening between the two arcs when constructing an equilateral triangle. Also, do not assume a triangle is equilateral just because it looks balanced; use equal lengths to justify the claim.
Problem
Construct ΔABC with every side 4 cm long.
- 1.Draw AB = 4 cm. This fixes the first required side.
- 2.Draw a 4 cm arc from A and another 4 cm arc from B. Call their intersection C.
- 3.Join AC and BC. Both are radii of their respective 4 cm arcs, so AC = BC = AB = 4 cm.
Problem
Construct an equilateral triangle with side length 3.5 cm.
- 1.Draw a base PQ = 3.5 cm.
- 2.Set the compass opening to 3.5 cm and draw intersecting arcs from P and Q. Name an intersection R.
- 3.Join PR and QR. The same distance argument gives PQ = PR = QR = 3.5 cm; decimal lengths do not change the method.
Problem
A learner draws AB = 4 cm, AC = 4 cm, and BC = 4.3 cm. Is the triangle equilateral?
- 1.Compare all three sides, not just two.
- 2.AB and AC are equal, but BC has a different length.
- 3.The triangle is not equilateral. Reposition C using two arcs with the same 4 cm opening rather than guessing its location.
Quiz
In ∠CAB, which point is the vertex of the angle?
Which measurements describe an equilateral triangle?
Why is the intersection C of two 4 cm arcs useful?
Which tool keeps the distance from a chosen centre fixed while drawing an arc?
What happens if the three chosen vertices lie on one straight line?
Practice Problems
- Draw ΔPQR and name all its sides and interior angles using three-letter angle names.
- Explain why ΔABC and ΔCAB can name the same triangle.
- Construct an equilateral triangle of side 5 cm and explain the purpose of each arc.
- Construct an equilateral triangle of side 3.5 cm. Verify all three sides with a ruler.
- Draw a 4 cm base and locate an equilateral third vertex on each side of the base. What is the same in both constructions?
- A learner draws a 5 cm base but uses 4 cm arcs from both endpoints. Explain why the result is not equilateral.
Key Takeaways
• A triangle has three non-collinear vertices, three sides, and three interior angles. • The middle letter in a three-letter angle name identifies its vertex. • An equilateral triangle has three equal sides. • A compass arc represents points at one fixed distance from its centre. • Two equal-radius arcs from the ends of an equally long base construct an equilateral triangle.
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Next · Lesson 2
Constructing a Triangle When its Sides are Given