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What is a Triangle?

  • A triangle is a polygon of three sides.
  • Triangles are classified in two general ways: by their sides and by their angles.

Types of Triangle

Triangles | CSAT Preparation - UPSC

Based on sides, triangles have been classified into three categories:

  1. Scalene Triangle: A triangle with three sides of different lengths is called a scalene triangle.
  2. Isosceles triangle: An isosceles triangle has two equal sides. The third side is called the base. The angles that are opposite to the equal sides are also equal.
  3. Equilateral triangle: An equilateral triangle has three equal sides. In this type of triangle, the angles are also equal, so it can also be called an equiangular triangle. Each angle of an equilateral triangle must measure 60o, since the sum of the interior angles of any triangle must equal to 180o.
    Triangles | CSAT Preparation - UPSC

Triangles | CSAT Preparation - UPSC          Triangles | CSAT Preparation - UPSC          Triangles | CSAT Preparation - UPSC

 

Triangles are also divided into three classes on the basis of measure of the interior angles:

  1. Obtuse Angled Triangle: When the measure of the largest angle of the triangle is greater than 900 then it is an obtuse angled triangle.
    In the figure ΔABC is an obtuse triangle where C is an obtuse angle.
    Triangles | CSAT Preparation - UPSC
  2. Acute Angled Triangle: In which all angles are less than 90o e.g.
    ΔPQR is an acute triangle because largest angle is less than 90o.
    Triangles | CSAT Preparation - UPSC
  3. Right Angled Triangle:
    A triangle whose one angle is 90o is called a right (angled) Triangle.
    In figure, b is the hypotenuse, and a & c the legs, called base and height respectively.
    In right triangle ABC we have,
    a2 + b2 = c2
    Triangles | CSAT Preparation - UPSC
    Triangles | CSAT Preparation - UPSC

Properties of Triangle

  1. Sum of the three angles is 180o.
  2. Sum of exterior angles of any triangle is equal to 360o.
  3. An exterior angle is equal to the sum of the interior opposite angles.
  4. The sum of the two sides is always greater than the third side.
  5. The difference between any two sides is always less than the third side.
  6. The side opposite to the greatest angle is the greatest side and the side opposite to the smallest angle will be the shortest side.
  7. If ΔABC is a right-angled triangle then ∠B is equal to 90o unless mentioned otherwise.
  8. In a right-angled triangle whose angles are 30o, 60o, 90o.
    In this triangle side opposite to angle 30o = Triangles | CSAT Preparation - UPSC Hypotenuse
    Side opposite to Angle 60o = Triangles | CSAT Preparation - UPSC Hypotenuse.
  9. Centroid:
    (a) The point of intersection of the medians of a triangle. (Median is the line joining the vertex to the
    mid-point of the opposite side).
    (b) The centroid divides each median from the vertex in the ratio 2 : 1.
    (c) To find the length of the median we use the theorem of Apollonius.
    AB2 + AC2 = 2(AD2 + BD2)
    Triangles | CSAT Preparation - UPSC
    (d) The medians will bisect the area of the triangle.
    (e) If x, y, z are the lengths of the medians through A, B, C of a triangle ABC, then
    “Four times the sum of the squares of medians is equal to three times the sum of the square of the sides of the triangle”.
    4(x2 + y2 + z2) = 3(a2 + b2 + c2).
  10. Orthocentre: This is the point of intersection of the altitudes. (Altitude is a perpendicular drawn from a vertex of a triangle to the opposite side.)
    In a right angled triangle, the orthocenter is the vertex, where the right angle is.
    Triangles | CSAT Preparation - UPSC
  11. Circumcentre: It is the point of intersection of perpendicular bisectors of the sides of the triangle.
    (a) The Circumcentre of a triangle is the centre of the circle passing through the vertices of a triangle.
    (b) The Circumcentre is equidistant from the vertices.
    (c) If a, b, c, are the sides of the triangle, Δ is the area, then abc = 4R Δ where R is the radius of the circum-circle.
    Triangles | CSAT Preparation - UPSC
  12. Incentre: This is the point of intersection of the internal bisectors of the angles of a triangle.
    Triangles | CSAT Preparation - UPSC
    Triangles | CSAT Preparation - UPSC
    (c) Δ = rs if r is the radius of incircle, where s = semi-perimeter = Triangles | CSAT Preparation - UPSC
    and Δ is the area of the triangle.
    (d) BF = BD = s – b where 2s = a + b + c,
    CE = CD = s – c
    AF = AD = s – a
    (e) The angle between the internal bisector and the external bisector is 90°.
    Triangles | CSAT Preparation - UPSC

Example.1 In ΔABC, AB = 9, BC = 10, AC = 12. Find the length of median through A.

  • In the adjacent figure AD is the required median. Using
  • Apollonius theorem in the triangle we have,
    2AD2 + 2(5)2 = 81+ 144 .
    2AD2 + 50 = 225
    Triangles | CSAT Preparation - UPSC
    Triangles | CSAT Preparation - UPSC

 

Example.2 The sides of the triangle are 6 cm, 8 cm, and 10 cm. Find the area, Inradius and Circumradius of the triangle.

  • s =  Triangles | CSAT Preparation - UPSC
    Triangles | CSAT Preparation - UPSC

 

➢ Equilateral Triangle

In an equilateral triangle, all the sides are equal and all the angles are equal.
(a) Altitude =  Triangles | CSAT Preparation - UPSC

(b) Area =  Triangles | CSAT Preparation - UPSC

(c) Inradius = Triangles | CSAT Preparation - UPSC

(d) Circumradius = Triangles | CSAT Preparation - UPSC

Triangles | CSAT Preparation - UPSC

Triangles | CSAT Preparation - UPSC

 

➢ Congruency

  • Two or more figures can be said congruent if and only if they all have same size and shape. If we talk about plane figures then they are congruent if their corresponding sides and angles are equal to the corresponding sides and angles of the other figure. 
  • Example: Two triangles ABC and DEF are said to the congruent, if they are
    equal in all respects (equal in shape and size).
    The notation for congruency is ≅ or ≡

➢ 

Congruent Triangles

  • If ∠A =∠D, ∠B = ∠E, ∠C = ∠F
  • AB = DE, BC = EF; AC = DF
  • Then ΔABC ≡ ΔDEF or ΔABC ≅ ΔDEF

Triangles | CSAT Preparation - UPSCTriangles | CSAT Preparation - UPSC

 

The Tests for Congruency:
(a) SAS Test: Two sides and the included angle of the first triangle are respectively equal to the two sides and included angle of the second triangle.
(b) SSS Test: Three sides of one triangle are respectively equal to the three sides of the other triangle.
(c) ASA Test: Two angles and one side of one triangle are respectively equal to the two angles and one side of the other triangle.
(d) RHS Test: The hypotenuse and one side of a right-angled triangle are respectively equal to the hypotenuse and one side of another right-angled triangle.

 

➢ 

Mid-point Theorem

  • A line joining the mid points of any two sides of a triangle must be parallel to the third side and equal to half of that (third side).
  • In the adjacent triangle ABC, if D and E are the respective mid-points of sides AB & AC, then DE II BC and DE =  Triangles | CSAT Preparation - UPSC  BC
    Triangles | CSAT Preparation - UPSC


➢ 

Similar Triangles

  • Two figures are said to be similar, if they have the same shape but not necessarily
    the same size. If two triangles are similar, the corresponding angles are equal
    and the corresponding sides are proportional.
  • In the figure:
    ΔABC ∼ ΔDEF then,
    ∠A = ∠D, ∠B = ∠E & ∠C = ∠F
    Triangles | CSAT Preparation - UPSC
    Triangles | CSAT Preparation - UPSCTriangles | CSAT Preparation - UPSC

Test for Similarity of Triangles:
(a) AAA Similarity Test: Three angles of one triangle are respectively equal to the three corresponding angles of the other triangle.
(b) SAS Similarity Test: The ratio of two corresponding sides is equal and the angles containing the sides are equal.
(c) SSS Similarity Test: The ratio of all the three corresponding side of the two triangles are equal.

➢ 

Basic Proportionality Theorem

  • In a triangle if a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the other two sides proportionally.
  • If in ΔABC, DE is drawn parallel to BC, it would divide sides AB and AC proportionally i.e.
    Triangles | CSAT Preparation - UPSC
    Triangles | CSAT Preparation - UPSC
    Triangles | CSAT Preparation - UPSC

Important Result

  • If in ΔABC DE II BC, and a line is drawn passing through A and parallel to BC.
    Then we will have:
    Triangles | CSAT Preparation - UPSC
    Triangles | CSAT Preparation - UPSC

➢ 

Areas of Similar Triangles

  • The ratios of the areas of two similar triangles are equal to the ratio of the square of their corresponding sides i.e. If ΔABC ∼ ΔDEF then
    Triangles | CSAT Preparation - UPSC
  • The ratio of the areas of two similar triangles is also equal to:
    (a) Ratio of the square of their corresponding medians.
    (b) Ratio of the square of their corresponding Altitudes.
    (c) Ratio of the square of their corresponding angle bisectors.
  • If two triangles are similar, the following properties are true:
    (a) The ratio of the medians is equal to the ratio of the corresponding sides.
    (b) The ratio of the altitudes is equal to the ratio of the corresponding sides.
    (c) The ratio of the internal bisectors is equal to the ratio of the corresponding sides.


➢ 

Angle Bisector Theorem

  • The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle. i.e. In a Δ ABC in which AD is the bisector of ∠A, then
    Triangles | CSAT Preparation - UPSC
    Triangles | CSAT Preparation - UPSC

 

➢ 

Intercept Theorem

  • Intercepts made by two transversals (cutting lines) on three or more parallel lines are proportional. 
  • In the figure, lines l and m are transversals to three parallel lines AB, CD, EF. Then, the intercepts (portions of lengths between two parallel lines) made, AC, BD & CE, DF, are respectively proportional.
    Triangles | CSAT Preparation - UPSC
    Triangles | CSAT Preparation - UPSC


➢ 

Pythagoras Theorem

  • The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides i.e. in a right angled triangle ABC, right angled at C,
    Triangles | CSAT Preparation - UPSC
  • Pythagorean Triplets:
    Pythagorean triplets are sets of 3. Integers which can be taken as the three sides
    of a right-angled triangle. Few Pythagorean triplets are (3, 4, 5), (5, 12, 13), (7, 24, 25), (9, 40, 41) etc.
    Triangles | CSAT Preparation - UPSC                                       
    Triangles | CSAT Preparation - UPSC


➢ 

Acute Angled Theorem

  • In an acute angle triangle ABC, AD is the altitude on BC from vertex A, and ∠ABC is the greatest angle among all the three angles. Then:
    Triangles | CSAT Preparation - UPSC

                       Triangles | CSAT Preparation - UPSC

 

➢ 

Obtuse Angled Theorem

  • In an obtuse angle triangle ABC, AD is the altitude on CB produced from vertex A, and ∠ABC is the greatest angle among all the three angles. 
  • Then AC2 = AB2 + BC2 + 2BD × BC
    Triangles | CSAT Preparation - UPSC

 

➢ 

Apollonius’ Theorem

  • This theorem is the combination of above two theorems and gives the length of the median. If in ΔABC, AD is the median, meeting side BC at D. ThenTriangles | CSAT Preparation - UPSC
    Triangles | CSAT Preparation - UPSC
The document Triangles | CSAT Preparation - UPSC is a part of the UPSC Course CSAT Preparation.
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FAQs on Triangles - CSAT Preparation - UPSC

1. What is a triangle?
A triangle is a polygon with three sides, three vertices, and three angles. It is one of the basic shapes in geometry.
2. What are the types of triangles?
There are several types of triangles based on their sides and angles. The common types include equilateral triangles (all sides and angles are equal), isosceles triangles (two sides and two angles are equal), scalene triangles (all sides and angles are different), acute triangles (all angles are less than 90 degrees), obtuse triangles (one angle is greater than 90 degrees), and right triangles (one angle is exactly 90 degrees).
3. What are some properties of triangles?
Some properties of triangles include the sum of all interior angles is always 180 degrees, the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles, the longest side of a triangle is opposite the largest angle, and the shortest side is opposite the smallest angle.
4. How do you find the area of a triangle?
The area of a triangle can be found using the formula: Area = (base * height) / 2. The base is the length of any side of the triangle, and the height is the perpendicular distance from the base to the opposite vertex.
5. Can a triangle have more than one right angle?
No, a triangle cannot have more than one right angle. In a right triangle, one angle measures exactly 90 degrees, and the sum of all angles in a triangle is always 180 degrees. Therefore, if one angle is 90 degrees, the sum of the other two angles must be 90 degrees as well, making it impossible for a triangle to have more than one right angle.
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