- Sum of the three
angles of a triangle is 180°
- The exterior
angle of a triangle (at each vertex) is equal to the sum of the two opposite
interior angles (exterior angle is the angle formed at any vertex, by une side
and the extended portion of the second side at that vertex. ∠Z = ∠X + ∠Y .
- An equilateral
triangle is one in which all the sides are equal (and hence, all angles are
equal, i.e., each of the angles is equal to 60°). An isosceles triangle is one in which two
sides are equal (and hence, the angles opposite them are equal). A scalene
triangle is one in which no two sides are equal .
- Sum of any two
sides of a triangle is greater than the third side; difference of any two sides
of a triangle is less than the third side. .
- If the sides are
arranged in the ascending order of their measurement, then the angles opposite
the sides (in the same order) will also be in ascending order (i.e., greater
angles has greater side opposite to it); if the sides are arranged in
descending order of their measurement, the angles opposite the sides in the
same order will also be in descending order (i.e., smaller angle has smaller
side opposite to it). .
- There can be
only one right angle or only one obtuse angle in any triangle. (Hence, in a
Right angled triangle, hypotenuse is the largest side). .
- In a right
angled triangle, the square on the hypotenuse (the side opposite the right
angle) is equal to the sum of the squares on the other two sides. In the figure
alongside, AC2 = AB2 + BC2 .
- In an obtuse
angled triangle, the square of the side opposite the obtuse angle is greater
than the sum of the squares of the other two sides by a quantity equal to twice
the product of one of the sides containing the obtuse angle and the projection
of the second side on the first side. In the adjacent figure, AC2 =
AB2 + BC2 + 2 BC.BD = AB2 + BC2 + 2
- In an acute
angled triangle, the square of the side opposite the acute angle is less than
the sum of the squares of the other two sides by a quantity equal to twice the
product of one of these two sides with projection of the second side on the
first side. In the figure alongside, AC2 = AB2 + BC2
- 2 BC.BD .
Geometric Points of a Triangle
The three perpendicular bisectors of a
triangle meet at a point called circumcentre of the triangle. The circumcentre
of a triangle is equidistant from its vertices and the distance of
circum-centre from each of the three vertices is called circumradius(R) of the
triangle. The circle drawn with the circumcentre as centre and circumradius as
radius is called the cirumcircle of the triangle and passes through all the
three vertices of the triangle. .
(B) Incentre & Excentres:
The (internal) bisectors of the three angles
of a triangle meet at a point called Incentre(I) of the triangle. Incentre is
equidistant from the three sides of the triangle i.e., the perpendicular’s
drawn from the incentre to the three sides are equal in length and this length
is called the inradius of the triangle. The circle drawn with incentre as
centre and inradius as radius is called the incircle of the triangle and it
touches all three sides on the inside. ∠BIC = 90 + 1/2 ∠A where I is the
incentre. Similarly, if the internal bisector of one angle and the external
bisectors of the other two angles are drawn, they meet at a point called
excentre. There will be totally three excentres for the triangle - one
corresponding to the internal bisector of each angle. .
The perpendicular drawn from a vertex to the
opposite side is called an altitude_ The three altitudes meet at a point called
Median is the line joining the mid-point of a
side to the opposite vertex. The three medians of a triangle meet at a point
called the centroid, G. .
Please note the following with reference to
the geometric centres of a triangle ABC. .
the angular bisector of angle A meets side BC at D, then BD/DC = AB/AC .
a right angled triangle, the vertex where the right angle is formed (i.e., the
vertex opposite the hypotenuse) is the orthocentre. .
an obtuse angled triangle, the orthocentre lies outside the triangle. .
R is the circum radius and r the inradius of the triangle, then the area of the
- = abc/4R = r.s where s = semiperimeter = (a + b + c)/2 .
divides each of the medians in the ratio 2 : 1, the part of the median towards
the vertex being twice in length to the part towards the side. .
A ABC, if AD is the median from A to side BC meeting BC at its mid point D,
then 2(AD2 + BD2) = AB2 + AC2. This
is called the Apollonius Theorem. .
a right angled triangle the length of the median drawn to the hypotenuse is
equal to half the hypotenuse. This median is also the circumradius and the
mid-point of the hypotenuse is the circumcentre. In an obtuse angled triangle,
the circumcentre lies outside the triangle. .
an isoceles triangle, the centroid, the orthocentre, the circumcentre and the
incentre, all lie on the median to the base. .
an equilateral triangle, the centroid, the ortho-centre, the circumcentre and
the incentre, all coincide. .
Congruency of Triangles
Two triangles that are identical in all
respects are said to be congruent. In two congruent triangles, .
corresponding sides (i.e., sides opposite to equal angles) are equal. .
corresponding angles (angles opposite to equal sides) are equal. .
Two triangles will be congruent if at least
one of the following conditions is satisfied:
sides of one triangle are respectively equal to the three sides of the second
sides and the included angle of one triangle are respectively equal to two
sides and the included angle of the second triangle. .
angles and one side of a triangle are respectively equal to two angles and the
corresponding side of the second triangle. .
right angled triangles are congruent if thc hypote-nuse and one side of one
triangle are respectively equal to the hypotenuse and one side of the second
right angled triangle. .
Similarity of Triangles
Two triangles are said to be similar if they
are alike in shape only. The corresponding angles of two similar triangles are
equal but the corresponding sides are only proportional and not equal. Two
triangles are similar if .
three angles of one are respectively equal to the three angles of the second
sides of triangle are proportional to two sides of the other and the included
angles are equal. .
Some more useful points about triangles
- A line drawn parallel to one side of a
triangle divides the other two sides in the same proportion. Conversely, a line
joining two points (each) dividing two sides of a triangle in the same ratio is
parallel to the third side. The ratio, the length of this line segment bears to
that of the third side is the same as that in which it cuts each of the first
two sides. .
- In two similar triangles,
- (a) Ratio of sides = Ratio of heights =
Ratio of medians = Ratio of angular bisectors = Ratio of in radii = Ratio of
circum radii. .
- (b) Ratio of area = Ratio of squares of
corresponding sides. .
- In a right angled triangle, the altitude
drawn from the vertex of Right Angle to the hypotenuse divides the given
triangle into two similar triangles, each of which is in turn similar to the
original triangle. .
- If ‘a’ is the side of an equilateral
triangle, Area of the triangle = /4a2
and the Height of the triangle = /2.a.