Construction of Triangles: quick revision, 15 bites

Constructing a Triangle from Three Sides

Three sides? Two arcs unlock the triangle.

To construct a triangle when all three side lengths are given, use **ruler and compass** to find the third vertex by intersecting two arcs.

Takeaway. The intersection of two arcs gives the exact location of the third vertex.

What is a Triangle?

Three non-collinear points make a triangle.

Takeaway. A triangle has six elements: three sides and three angles.

Constructing an Equilateral Triangle

Two circles, one perfect triangle.

An equilateral triangle has all sides equal. Use compass to draw two circles of the same radius from each endpoint of the base.

Takeaway. The intersection of two equal-radius circles gives the third vertex of an equilateral triangle.

Triangle Inequality Theorem

Any two sides beat the third.

The sum of the lengths of any two sides of a triangle is **always greater** than the length of the third side.

Takeaway. If the sum of two sides is not greater than the third, no triangle can be formed.

Don't Ignore the Triangle Inequality

2+3=5? No triangle with 6!

Takeaway. Check the triangle inequality before starting any construction.

Range of the Third Side

Third side: between difference and sum.

If two sides of a triangle are given, the third side must lie between the difference and the sum of the two sides.

Takeaway. The third side is always greater than the difference and less than the sum of the other two.

Altitude of a Triangle

Perpendicular from vertex to opposite side.

Takeaway. An altitude is a perpendicular segment from a vertex to the line containing the opposite side.

Constructing an Altitude Using Compass

Arc, intersect, draw perpendicular.

Takeaway. The altitude is constructed by finding the perpendicular from the vertex to the opposite side.

Orthocentre

Three altitudes meet at one point.

The three altitudes of a triangle are **concurrent**; they meet at a single point called the **orthocentre**.

Takeaway. The point where all altitudes intersect is the orthocentre.

Classifying Triangles by Sides

Equilateral, isosceles, or scalene?

Scalene: all sides different.

Takeaway. Triangles are classified by the number of equal sides.

Classifying Triangles by Angles

Acute, right, or obtuse?

Obtuse: one angle > 90°.

Takeaway. Triangles are classified by their largest angle.

Angle Sum Property

All angles add to 180°.

Takeaway. The sum of interior angles of any triangle is always 180°.

Only One Right or Obtuse Angle

Two right angles? Impossible!

Takeaway. Due to the angle sum property, only one angle can be ≥ 90°.

Finding the Third Angle

42° + 17° = 59°, so third is 121°.

Takeaway. Use the angle sum property to find a missing angle.

Triangles in Architecture

Bridges and buildings rely on triangles.

Triangles are used in bridges, roofs, and cranes because they are **rigid** and cannot be deformed without changing side lengths or breaking joints.

Takeaway. The rigidity of triangles makes them essential in construction.