Triangles – Formula & Equation Sheet
Essential formulas and equations from Mathematic, tailored for Class 10 in Mathematics.
This one-pager compiles key formulas and equations from the Triangles chapter of Mathematic. 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
Area of Triangle: A = 1/2 × base × height
A is the area (in square units), base is the length of the base of the triangle, and height is the perpendicular distance from the base to the opposite vertex. This formula is fundamental for calculating the area of triangles.
Pythagorean Theorem: a² + b² = c²
a and b are the lengths of the legs of a right triangle, and c is the length of the hypotenuse. This theorem is used to relate the sides of right triangles, crucial for solving geometry problems.
Congruent Triangles: △ABC ≅ △DEF
This notation indicates that triangle ABC is congruent to triangle DEF, meaning they have identical sizes and shapes. It forms the basis for similarity comparisons.
Similarity of Triangles: △ABC ~ △DEF
This notation indicates that triangle ABC is similar to triangle DEF, meaning their corresponding angles are equal and their sides are in proportion.
Basic Proportionality Theorem: AD/DB = AE/EC
For line segment DE parallel to BC in triangle ABC, where D and E are points on sides AB and AC respectively. This theorem shows the proportional relationship between the divided segments.
AAA Similarity Criterion: If ∠A = ∠D, ∠B = ∠E, ∠C = ∠F, then △ABC ~ △DEF
This criterion states that if all corresponding angles of two triangles are equal, then the triangles are similar.
SSS Similarity Criterion: If AB/DE = AC/DF, then △ABC ~ △DEF
This criterion states that if the sides of two triangles are in proportion, the corresponding angles are equal, and thus the triangles are similar.
SAS Similarity Criterion: If ∠A = ∠D and AB/DE = AC/DF, then △ABC ~ △DEF
This states that if one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in proportion, the triangles are similar.
Height of Triangle: h = (2A)/base
h is the height of the triangle, A is the area, and base is the base length. This is useful for finding the height when the area and base are known.
Ratio of Areas of Similar Triangles: Area1/Area2 = (side1/side2)²
This formula indicates that the ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding sides, helping to solve area-related problems.
Equations
AD/DB = AE/EC (Basic Proportionality)
This fundamental relationship arises when a line is drawn parallel to one side of a triangle, leading to proportional divisions of the other two sides.
A = 1/2 × b × h (Area of Triangle)
This equation gives the area of a triangle where b is the base and h is the height, essential for calculating triangles in geometric problems.
a² + b² = c² (Pythagorean Theorem)
This classic equation relates the sides of right triangles and is vital in determining unknown lengths given certain conditions.
If two triangles are similar, then: AB/DE = BC/EF = AC/DF
This equation expresses proportionality between corresponding sides of similar triangles, a key principle in triangle similarity.
∠A + ∠B + ∠C = 180° (Sum of Angles in Triangle)
This equation states that the sum of interior angles in any triangle equals 180 degrees, crucial for solving angle-related problems.
If AD/DB = AE/EC, then DE || BC (Converse of Basic Proportionality)
Illustrates that if a line divides two sides of a triangle proportionally, it is parallel to the third side, important for proving similarities.
If △ABC ~ △DEF, then A1/A2 = (s1/s2)²
This equation relates the areas of two similar triangles to the squares of their corresponding sides, used in area calculations.
A = bh (Area of Triangle)
Gives the area of a triangle; b = base, h = height, essential in solving problems involving triangle areas.
AD/DB = EC/AE (If DE || BC)
This states that the segments created by a line parallel to one side of a triangle create equal ratios with the other sides.
h = (2A)/base
Defines the height in terms of area and base, helping to compute one variable when the other two are known.
