Triangles - Kamal Lohia
Faculty and Content Developer at Tathagat | Delhi College of Engineering
According to lengths of the side, we divide triangles in three categories.
Equilateral – all sides are equal.
Isosceles – at least two sides are equal.
Scalene – all three sides are different.
But can we form a triangle with any three sides?
Let’s talk with an example:
What is the area of the triangle whose three side lengths are 1, 2 and 3 units respectively?
Triangle is not possible. Or, mathematically, we should say area of triangle is zero.
So question arises, what is the condition on lengths of sides of a triangle to confirm that a triangle is possible.
Just lay down the longest side as base of the triangle. Now remaining two sides should not overlap on base. So, for this reason, sum of those other two side lengths of the triangle should be greater than longest side of the triangle.
Triangle Inequality: If a, b, c are the side lengths of a triangle then sum of any two sides of a triangle is greater than third side i.e. a + b > c, b + c > a and c + a > b
Also difference of two sides is always less than third side of triangle
i.e. a – b < c, b – c < a and c – a < b
I know that you know almost all of these formulae already. But target here is to analyze them more deeply. If I ask you under what conditions the area of the triangle becomes maximum, can you think a way out?
Some special lines and points in triangles
MEDIAN & CENTROID
A line segment joining mid-point of a side of a triangle with opposite vertex is known as MEDIAN.
There are three medians possible in every triangle.
Median divides a triangle in two equal area triangles.
All the three medians intersect at a point, known as CENTROID.
CENTROID divides every median in the ratio 2 : 1
All the three medians divided the triangle in six equal area triangles.
CENTROID always lies inside a triangle.
ANGLE BISECTORS & INCENTER
Each angle bisector of the triangle divides the opposite side in the ratio of the sides containing the angle. (Angle Bisector Theorem)
All three angle bisectors meet at a point which is known as INCENTER i.e. center of inscribed circle.
INCENTER is a point inside the triangle which is equidistant from three sides.
INCENTER is center of inscribed circle of the triangle which touches all the three sides of the triangle.
INCENTER always lies inside a triangle.
PERPENDICULAR BISECTOR & CIRCUMCENTER
Perpendicular bisector is a line which divides a side of the triangle in two equal halves perpendicularly.
All the three perpendicular bisectors of a triangle intersect at a common point which is known as CIRCUMCENTER.
CIRCUMCENTER is a point which is equidistant from three vertices of the triangle.
CIRCUMCENTER is center of circumscribed circle of the triangle which passes through all the three vertices.
For acute triangle, circumcenter lies inside the triangle.
For right triangle, circumcenter lies at mid-point of hypotenuse.
For obtuse triangle, circumcenter lies outside the triangle.
ALTITUDES & ORTHOCENTER
Perpendicular dropped from a vertex to opposite side is known as an ALTITUDE.
All the three altitudes meet at a common point known as ORTHOCENTER.
Some special triangles
All three sides are same in length (say a).
Each angle is equal and equal to 60o.
All the four lines and points discussed above are same points in an equilateral triangle.
Height of an equilateral triangle is √3 a/2
Area of an equilateral triangle is √3 a2/4
Ratio of inradius to circumradius i.e. r/R = 1/2.
Pythagoras Theorem states a2 + b2 = c2 where c is the hypotenuse and a, b are legs.
There are two special cases of right triangles: 30-60-90 and 45-45-90 as shown. Note the ratio of their sides.
The two triangles, whose any three corresponding parameters (out of three sides and three angles) are identical are same to be congruent.
There are three exceptions to this rule and they are AAA, SSA and ASS.
Two congruent triangles are identical in all respects.
The two triangles are said to be similar
(I) if two of the angles of triangles are same – AA, or
(II) if two sides of a triangle are proportional to two sides of another triangle and angle included between the two sides in both the triangles is same – SAS.
Sides opposite to equal sides are said to be proportional sides and ratio of proportional sides is same. Area of two similar triangles is in the ratio of square of ratio of corresponding sides.
INRADIUS & CIRCUMRADIUS
Inradius (r) is radius of the incircle inscribed in a triangle which touches all the sides of the triangle.
Circumradius (R) is radius of circumcircle passing through all the vertices of triangle.
We have already mentioned the formulae to find r and R in terms of area of triangle.
If b, c are two sides of triangle and h is the altitude drawn from vertex common to b and c, then R = bc/2h.
For a right triangle with legs a, b and hypotenuse c, inradius r = (a + b – c)/2