Properties of Triangles: Theorems for CMAT
Understanding Triangles
Triangles are a fundamental component of CMAT Geometry. Questions usually focus on angles, side lengths, and area calculation. Understanding the relationship between these elements is key to solving problems efficiently without performing lengthy calculations.
Fundamental Theorems
- Angle Sum Property: The sum of all interior angles of a triangle is always $180^\circ$. For a triangle with angles $A, B,$ and $C$, $A + B + C = 180^\circ$.
- Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If the sides are $a, b, c$, then $a + b > c$, $a + c > b$, and $b + c > a$.
- Exterior Angle Theorem: The exterior angle of a triangle is equal to the sum of the two opposite interior angles. If an angle is extended, the exterior angle = sum of opposite interior angles.
Special Triangles
- Right-Angled Triangle: Follows the Pythagorean theorem: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse. Area is $\frac{1}{2} \times \text{base} \times \text{height}$.
- Equilateral Triangle: All sides are equal ($a=b=c$) and all angles are $60^\circ$. The area formula is $\frac{\sqrt{3}}{4} a^2$.
- Isosceles Triangle: Two sides are equal, and the angles opposite these sides are also equal.
Area Formulas
- Basic Formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.
- Heron's Formula: Used when three sides ($a, b, c$) are known. First, calculate the semi-perimeter: $s = \frac{a+b+c}{2}$. Then, $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$.
Comparison: Congruence vs Similarity
| Feature | Congruent Triangles | Similar Triangles |
|---|---|---|
| Definition | Identical in shape and size | Identical in shape, different size |
| Angles | Corresponding angles are equal | Corresponding angles are equal |
| Sides | Corresponding sides are equal | Corresponding sides are proportional |
| Notation | $\cong$ | $\sim$ |
Common Traps
- Mixing units: Ensure all dimensions are in the same unit (e.g., cm to m) before calculating area.
- Similarity Ratio: If the ratio of sides of two similar triangles is $k$, the ratio of their areas is $k^2$.
- Triangle Inequality: Always verify if the given three side lengths can actually form a triangle using the $a+b > c$ rule.
Practise these as timed MCQs to lock the pattern in.