Explore the properties, types, and theorems related to triangles, including congruence and similarity, to solve geometric problems effectively.
Triangles – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class X in Mathematics.
This one-pager compiles key formulas and equations from the Triangles chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
Basic Proportionality Theorem (Thales Theorem): If DE || BC, then AD/DB = AE/EC
AD, DB, AE, EC are segments of sides AB and AC. This theorem helps in finding unknown lengths in triangles when a line is parallel to one side.
Pythagoras Theorem: AB² + BC² = AC²
AB and BC are the legs, AC is the hypotenuse of a right-angled triangle ABC. Used to find the length of a side in a right-angled triangle.
Area of Triangle: ½ × base × height
Base and height are perpendicular to each other. Fundamental for calculating area in various problems.
AAA Similarity Criterion: If ∠A = ∠D, ∠B = ∠E, ∠C = ∠F, then ΔABC ~ ΔDEF
Triangles are similar if corresponding angles are equal. Useful for proving similarity without side lengths.
SSS Similarity Criterion: If AB/DE = BC/EF = CA/FD, then ΔABC ~ ΔDEF
Triangles are similar if all corresponding sides are in the same ratio. Essential for similarity proofs.
SAS Similarity Criterion: If AB/DE = AC/DF and ∠A = ∠D, then ΔABC ~ ΔDEF
Triangles are similar if two sides are in the same ratio and the included angles are equal. Key for quick similarity checks.
Ratio of Areas of Similar Triangles: (Area of ΔABC)/(Area of ΔDEF) = (AB/DE)²
AB and DE are corresponding sides. Shows area ratio is the square of the side ratio. Useful in geometry problems.
Mid-point Theorem: If D and E are mid-points of AB and AC, then DE || BC and DE = ½ BC
D and E divide AB and AC into two equal parts. Helps in finding parallel lines and segment lengths.
Converse of Mid-point Theorem: If D is mid-point of AB and DE || BC, then E is mid-point of AC
Extends the mid-point theorem to prove mid-points using parallel lines. Useful in construction problems.
Angle Bisector Theorem: AD/DB = AC/BC
AD is the angle bisector of ∠A in ΔABC. Helps in finding segment lengths when an angle is bisected.
Equations
For similar triangles ΔABC ~ ΔDEF, AB/DE = BC/EF = AC/DF
AB, BC, AC and DE, EF, DF are corresponding sides. Fundamental for solving problems involving similar triangles.
In right ΔABC, right-angled at B, sin A = BC/AC
BC is opposite side, AC is hypotenuse. Defines sine of angle A in terms of sides.
In right ΔABC, right-angled at B, cos A = AB/AC
AB is adjacent side, AC is hypotenuse. Defines cosine of angle A in terms of sides.
In right ΔABC, right-angled at B, tan A = BC/AB
BC is opposite side, AB is adjacent side. Defines tangent of angle A in terms of sides.
If ΔABC ~ ΔDEF, then ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
Corresponding angles of similar triangles are equal. Essential for angle-related proofs.
For equilateral triangles, all sides are equal and all angles are 60°
Simplifies problems by providing uniform properties. Useful in symmetry and area calculations.
In ΔABC, if ∠A = ∠B, then sides opposite to them are equal, i.e., BC = AC
Isosceles triangle property. Helps in identifying equal sides based on angles.
For two triangles to be congruent, corresponding sides and angles must be equal
Congruency criterion. Fundamental for proving exact match in shape and size.
In ΔABC, sum of angles ∠A + ∠B + ∠C = 180°
Angle sum property. Used to find missing angles in a triangle.
Exterior angle of ΔABC equals sum of opposite interior angles, i.e., ∠ACD = ∠A + ∠B
Exterior angle theorem. Useful in solving problems involving exterior angles.
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