A Tale of Three Intersecting Lines – Formula & Equation Sheet
Essential formulas and equations from Ganita Prakash, tailored for Class 7 in Mathematics.
This one-pager compiles key formulas and equations from the A Tale of Three Intersecting Lines chapter of Ganita Prakash. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formulas
Sum of Angles in a Triangle: ∠A + ∠B + ∠C = 180°
Where ∠A, ∠B, and ∠C are the angles of the triangle. This formula shows that the total angle measure in any triangle equals 180 degrees, which is fundamental in geometry.
Triangle Inequality Theorem: a + b > c, a + c > b, b + c > a
For sides a, b, and c of a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This helps verify if a triangle can be formed with given lengths.
Area of a Triangle: A = 1/2 * base * height
A is the area, the base is one side of the triangle, and the height is the perpendicular distance to that base. This formula is crucial for calculating the area of triangular shapes in real life.
Perimeter of a Triangle: P = a + b + c
P is the perimeter, while a, b, and c are the side lengths. This helps determine the total length around the triangle.
Median of a Triangle: Length = (1/2) * sqrt(2b² + 2c² - a²)
This formula calculates the length of a median from vertex A to side BC, where a, b, and c are the sides of the triangle. Medians help in various geometric analyses.
Altitude of a Triangle: h = (2A)/base
h is the height, A is the area, and base is one side of the triangle. This helps in finding the height if the area and base are known.
Pythagorean Theorem (for Right Triangles): a² + b² = c²
a and b are the lengths of the legs, and c is the length of the hypotenuse. This theorem is instrumental in identifying right triangles and calculating side lengths.
Equilateral Triangle: Each angle = 60°
In an equilateral triangle where all sides are equal, all angles measure 60 degrees. This is useful for dividing angles in geometric problems.
Isosceles Triangle: Base angles are equal
In isosceles triangles, the angles opposite to the equal sides are equal. This property is vital in angle calculations.
Scalene Triangle: All sides and angles are different
Scalene triangles have no equal sides or angles, which signifies unique properties in angular calculations.
Equations
AB + AC > BC
Where AB, AC, and BC are the lengths of the triangle's sides. This follows the Triangle Inequality Theorem, ensuring triangle formation.
AC + BC > AB
This inequality further supports the Triangle Inequality Theorem concept, essential for validating side lengths for triangle construction.
BA + AC > BC
Another representation of the Triangle Inequality Theorem. It ensures stability in triangular structures or designs.
h = sqrt(b² - (a/2)²)
For an isosceles triangle, where h is the height from the vertex to the base, b is the base length, and a is the length of the equal sides. This aids in calculating heights based on known lengths.
Area (Equilateral Triangle) = (sqrt(3)/4) * a²
Where a is the length of a side. This specific formula is beneficial for finding the area of equilateral triangles directly.
A = (1/2) * a * h
This is another representation of the area formula, emphasizing base height combinations for calculating triangle areas.
P = (x + y + z)
Where x, y, and z are the sides of scalene triangles. This formula confirms the fundamental perimeter calculation.
Angle Sum = 180°
This equation confirms that in any triangle, the sum of angles equals 180°, pivotal for angle evaluations.
a < b + c
It expresses the condition indicating that no side of a triangle can be equal to or longer than the sum of the other two sides, vital in geometrical validation.
A = 1/2 * a * b * sin(C)
This is the area formula using two sides and the included angle C. This helps calculate the area when specific angle measures and side lengths are known.
