Triangles

Sum of the three angles of a triangle is 180°
Fig.3.
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. ÐZ = ÐX + ÐY .
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 .

Fig.4.

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 AB.BE .

Fig.5.

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

(A) Circumcentre:

Fig.6.

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:

Fig.7.

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. ÐBIC = 90 + 1/2 ÐA 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. .

(C) Orthocentre:

Fig.8.

The perpendicular drawn from a vertex to the opposite side is called an altitude_ The three altitudes meet at a point called orthocentre. .

(D) Centroid:

Fig.9.

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. .

NOTE:

Please note the following with reference to the geometric centres of a triangle ABC. .

I. If the angular bisector of angle A meets side BC at D, then BD/DC = AB/AC .

II. In a right angled triangle, the vertex where the right angle is formed (i.e., the vertex opposite the hypotenuse) is the orthocentre. .

III. In an obtuse angled triangle, the orthocentre lies outside the triangle. .

IV. If R is the circum radius and r the inradius of the triangle, then the area of the triangle .

- = abc/4R = r.s where s = semiperimeter = (a + b + c)/2 .

V. Centroid 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. .

VI. In 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. .

VII. In 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. .

VIII. In an isoceles triangle, the centroid, the orthocentre, the circumcentre and the incentre, all lie on the median to the base. .

IX. In 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, .

(A) The corresponding sides (i.e., sides opposite to equal angles) are equal. .

(B) The corresponding angles (angles opposite to equal sides) are equal. .

Two triangles will be congruent if at least one of the following conditions is satisfied:

I. Three sides of one triangle are respectively equal to the three sides of the second triangle. .

II. Two sides and the included angle of one triangle are respectively equal to two sides and the included angle of the second triangle. .

III. Two angles and one side of a triangle are respectively equal to two angles and the corresponding side of the second triangle. .

IV. Two 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 .

(A) The three angles of one are respectively equal to the three angles of the second triangle

(B) Two 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.

Basic Concepts Of Plane Geometry Part 4