Trigonometric - Properties of Triangles (Lecture-04)
Properties and Solutions of Triangles
Notation :
Vertices A,B,C
Sides
Centroid G (Point of intersection of medians )
Orthocentre O (Point of intersection of altitudes)
Circumcentre S (Point of intersection of perpendicular bisectors of the sides)
Incentre I (Point of intersection of internal bisectors of the angles)
Excentres
Circumradius
Inradius
Exradii
Semiperimeter
Area of triangle
Concepts and Formula
1. Sine law
In any triangle
2. Cosine law
In any triangle
or or or
3. Projection formula
4. Napier’s analogy (Tangent rule)
5. Auxiliary formulae
(Trigonometrical ratios of half angles of a triangle)
6. Area of Triangle
-
Area of triangle
(Hero’s formula) -
-
7.
Also
8. Rule
In any triangle,

9. Circumcircle of a triangle.
Circle passing through the angular point of a
triangle. In a right angled triangle the circum centre is the mid-point of hypotenuse.
10. Incircle of a triangle
The circle which touches the sides is called inscribed circle or incircle. Its radius is denoted by r.
11. Escribed circles of a triangle.
The circle which touches side
12. Orthocentre and Pedal triangle
-
The triangle MNP formed by joining the feet of the altitudes is called the pedal triangle
-
The distance of orthocentre
from vertices are and -
Distance of
from sides are and . -
,

i.e. sides of pedal triangles are a
- Circumradii of the triangles
are equal.
13. Note :
-
Orthocentre of
is the incentre of pedal triangle MNP. -
Incentre I of
is the orthocentre of . -
Centroid of
lies on the line joining the circumcentre to the orthocentre and divides it in the ratio -
Circumcentre of the pedal triangle bisects the line joining the circumcentre of the triangle to the orthocentre.
14. Excentral Triangle.
The triangle formed by joining the three excentres
So clearly

-
Sides of excentral trangle are
and and its angles are -
15. Nine point circle
Circle circumscribing the pedal triangle of a given triangle bisects the sides of the given triangle and also the line joining the vertices of the given triangle to the orthocentre of the given triangle. This circle is known as nine point circle.
i.e. Nine point circle passes through the mid point of the sides, feet of the perpendiculars and the mid points of the line joining the orthocentre to the angular points.

16. Distance between special points
- Distance between excentre and circumcentre
-
Distance between cirumcentre \& orthocentre is
-
Distance between cirumcentre \& incentre is
-
Distance between incentre \& orthocentre is
-
Perimeter
and area of a regular polygon of sides inscribed in a circle of radius are given by and . -
Perimeter
and area of a regular polygon of sides circumscribed about a given circle of radius are given by and
17. Length of medians (Apollonious rule)
Also
If
Area
18. Altitudes
Area is given by
19. Length of internal bisectors of angles A,B,C are given by
20. Some useful results.
-
-
. -
-
-
-
-
-
-
-
21. Solution of triangles
(a) Solution of a general triangle.

(b) Solution of a right angled triangle
Let
Given | To find | Formulae used |
---|---|---|
a,b | A, B, c | |
(two sides) | ||
c,a | A, B, b | |
(hypotenuse & one side) | ||
a, A | B, b, c | |
(Side and angle) | ||
c,A | B, a, b | |
(hypotenuse & angle) | ||
Solved Examples
1. If in
(a) 8
(b) 9
(c) 6
(d) None of these
Show Answer
Solution: Using Napier’s analogy,
Answer (b)
2. If the angles
(a) isosceles
(b) equilateral
(c) acute angled
(d) None of these
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Solution:
But
Answer (d)
3. In
(a)
(b)
(c)
(d) None of these
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Solution: Given equation is,
Answer (a)
4. If the median of a triangle through
(a)
(b)
(c)
(d)
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Solution:

Answer (d)
5. In
(a)
(b)
(c)
(d)
Show Answer
Solution:

Area of
Answer (d)
6. In
(a) equilateral
(b) isosceles
(c) right angled
(d) None of these
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Solution:
Answer (c)
7. In
Show Answer
Solution:
In
Let
given that
Squaring,
i.e.
Exercise
1. In
(a)
(b)
(c)
(d)
Show Answer
Answer: a2. If in
(a) The altitudes are in A.P
(b) The medians are in G.P
(c) The altitudes are in H.P
(b) The medians are in A.P
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Answer: c3. In
(a)
(b)
(c)
(d)
Show Answer
Answer: a4. In
(a)
(b)
(c)
(d)
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Answer: b5. In
(a)
(b)
(c)
(d)
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Answer: a6. Inradius of a circle which is inscribed in an isosceles triangle one of whose angle is
(a)
(b)
(c)
(d) None of these
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Answer: c7. If the angles
(a)
(b)
(c) 1
(d)
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Answer: d8. Read the following passage and answer the questions.
Consider the circle
(i) The ratio of the areas of
(a)
(b)
(c)
(d)
(ii) The radius of the circumcircle of the
(a) 5
(b)
(c)
(d)
(iii) The radius of the incircle of the triangle
(a) 4
(b) 3
(c)
(d) 2
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Answer: (i) c (ii) b (ii) d9.* Internal bisector
(a)
(b)
(c)
(d)
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Answer: a,b,c,d10.* A straight line through the vertex
(a)
(b)
(c)
(d)
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Answer: b,d11.* In a
(a)
(b)
(c) locus of point
(d) locus of point
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Answer: b,c12. Match the following:
Column I | Column II | ||
---|---|---|---|
(a) | In |
(p) | |
(b) | In |
(q) | (q) |
(c) | Two sides of a triangle are given by the roots of the equation |
(r) | |
(s) |
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Answer:13. In
(a)
(b)
(c)
(d)
Show Answer
Answer: a14. In
(a)
(b)
(c)
(d)
Show Answer
Answer: b15. If
(a)
(b)
(c)
(d) None of these