Explore different types of triangles in geometry, their essential properties, perimeter and area formulas, and key trigonometric laws used to calculate angles and side lengths.
Angles and Perimeter of a Triangle
A triangle is a closed two-dimensional figure with three sides and three interior angles: \(\angle A, \angle B, \angle C\).
Important Note: The sum of all interior angles of a triangle is always \(180^\circ\):
\[ \angle A + \angle B + \angle C = 180^\circ \]Perimeter: The perimeter of a triangle is given by the sum of its three side lengths:
\[ P = a + b + c \]Area of a Triangle
The area \(A\) of a triangle can be found using different formulas depending on the known information:
- Base and height formula: \(A = \dfrac{1}{2} b h\)
- Heron's formula (when all sides are known): \(A = \sqrt{s(s-a)(s-b)(s-c)}\), where \(s = \dfrac{a+b+c}{2}\) is the semi-perimeter.
- Using sine: \(A = \dfrac{1}{2} a b \sin C\)
Types of Triangles & Properties
Isosceles Triangle
An isosceles triangle has two equal sides and two equal angles.
- Two equal sides: \(a = b\)
- Two equal angles: \(\angle A = \angle B\)
- The altitude, median, and angle bisector from the third vertex coincide.
Equilateral Triangle
An equilateral triangle has all three sides and angles equal.
- All sides are equal: \(a = b = c\)
- All angles are equal: \(\angle A = \angle B = \angle C = 60^\circ\)
Right Triangle & Pythagorean Theorem
A right triangle has one right angle (\(90^\circ\)). The longest side opposite the right angle is the hypotenuse.
Pythagorean Theorem:
\[ c^2 = a^2 + b^2 \]Sine Law and Cosine Law
Sine Law
The Sine Law helps solve non-right triangles:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]Cosine Law
The Cosine Law is useful for solving non-right triangles when side lengths or included angles are known:
\[ a^2 = b^2 + c^2 - 2bc \cos A \]