Quant (Concepts): Triangles

Quant (Concepts): Triangles

For Any triangle:

• b < a + c and b > |a – c|, where a, b & c are the three sides of the triangle.
• Sum of the angles of a triangle is 180^o.
• The length of a side is proportional to the sine of the angle opposite it. e.g. c∝ sin 

A median, which is a line connecting the midpoints of two sides is parallel to the third side and its length is equal to a half of the length of the third side.

Area = 1/2 × b × h

Area =  1/2 × b × a sin 

,
where S = 1/2 × (a + b + c)

Right-Angled Triangles:

• c ² = a ² + b ²

Area = 1/2 × a × b

• Radius of Circum-circle = 1/2 × c
• Radius of In-circle = 2 × Area/perimeter

Altitude from side c = ab/c

• In a right-angled isosceles triangle, if a = b, then, hypotenuse c = a √2, and Area =1/2a²

Equilateral Triangles:

For an equilateral triangle with side a,

• Altitude = √3/2 × a, Area = √3/4 × a ²
• Radius of Circum-circle = a/√3
• Radius of In-circle = a/2√3

Other Important Properties:

• The centroid of a triangle divides the medians in the ratio 2 : 1.
• If a, b and c are the lengths of the 3 sides of a triangle, then the in-radius, r = ^A/_S, where A is the area of the triangle and s is the semi-perimeter.
• Circum-radius, R = a × b × c/4A.
• The length of the circum-radius of a right-angled triangle is ¹/₂ × hypotenuse.
• In a triangle ABC, if D is a point on BC, such that AD is the angle bisector of ∠A, then AB/AC = BD/DC. (Angle bisector theorem)
• In a triangle ABC, if D is a point on BC, such that AD is the median, then AB ² + AC ² = 2(AD ² + DC ²). (Appollonius theorem)
• In a triangle ABC, the segment passing through the midpoints of AB and AC is parallel to BC and is 1/2 × BC.
• The measure of the exterior angle is the sum of the remote interior angles in the triangle.
• In a right-angled triangle ABC, if D is a point on BC, such that AD is the median, then AD =1/2 × BC (B is the right angle).
• Of all the triangles with the same perimeter, the triangle with the maximum area is the equilateral triangle.
• In a triangle ABC, if D is a point on BC, such that AD is perpendicular to the hypotenuse, BC, then AD = AB × AC/BC.
• The areas of 2 similar triangles are proportional to the squares of their corresponding sides.