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Triangles

Triangles

A triangle is a three sided shape with three angles, and the angles add up to 180°.

Equlilateral Triangle The angles are equal as are the length of the sides.

Isosceles Triangle Two sides are equal and two angles are equal

Scalene Triangle The sides have different length so no angles are equal.

Right Angled Triangle One of the angles is a right angle.

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The Cosine Rule

Can be used for all triangles, not just right angled triangles. The cosine rule formula is derived by dividing a scalene triangle into two right angled triangles.

The Sine Rule

Can be used for all triangles, not just right angled triangles. The sine rule formula is derived by dividing a scalene triangle into two right angled triangles.

Sum of Angles

Sum of Angles

The angles add up to 180°; this is a property of physical reality there is no actual proof, it is however it is self evident. We can draw any triangle, and make two copies, and then align the angles as shown in the animation. In all cases the angles form a straight line.

Triangle Area Formula

The area A of any triangle is:

$$ A=\frac{1}{2} bh $$

where b is the length of the base and h is the height of triangle. We can prove this formula by recalling the formula for the area \( A_{r}=wh \) of a rectangle which is product of the width of the base \( w \) multiplied by the height \( h \). This formula is intrinsic to 3D space and there is no proof. We can draw one diagonal to define two identical right angled triangles, and each triangle has half the area of the rectangle.

We can draw any triangle (shown in yellow). A line is drawn from B which is perpendicular to AC. This line intersects the line through AC at D.

When angle C is obtuse: The area \( A \) of △ ABC is the difference in area of two right angled triangles, namely △ ADB minus the △ CDB. $$ A= \frac{1}{2}((b+x)h-xh)$$ expanding brackets $$ A= \frac{1}{2}((bh+xh-xh)$$ and simplifying: $$ A=\frac{1}{2} bh $$

When angle C is acute: The area \( A \) of △ ABC is the sum of the area of two right angled triangles, namely △ ADB the △ DBC. $$ A= \frac{1}{2}((b-x)h+xh)$$ expanding brackets $$ A= \frac{1}{2}((bh-xh-xh)$$ and simplifying: $$ A=\frac{1}{2} bh $$ The formula has now been proved for all triangles.

Discussion

By drawing a perpendicular line from a vertice of a triangle, we can divide it into two right angled triangles. This property of triangles is used to prove the triangle area formula.