Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

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Triangles

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
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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev          Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev          Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

 

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


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


Acute angled triangle:
In which all angles are less than 90o e.g.
ΔPQR is a acute triangle because largest angle is less than 90o.

Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

 

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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRevTriangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRevTriangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

 

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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev Hypotenuse


Side opposite to Angle 60o = Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev 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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

(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).

 

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

Sol. 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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

 

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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

 

12. Incentre:
This is the point of intersection of the internal bisectors of the angles of a triangle.
Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev
(c) Δ = rs if r is the radius of incircle, where s = semi-perimeter = Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev
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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

 

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


Sol. s =  Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev
Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

 

Equilateral triangle
In an equilateral triangle all the sides are equal and all the angles are equal.
(a) Altitude =  Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

(b) Area =  Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

(c) Inradius = Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev


(d) Circumradius = Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

 

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. E.g.:- 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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRevTriangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

 

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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev  BC

Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

 

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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

 

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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

 

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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

 

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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

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.


Properties

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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

 

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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

 

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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev


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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev                                        Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

 

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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev                       Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

 

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 - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

 

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. Then

Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

Triangles - Examples (with Solutions), Geometry, Quantitative Reasoning Quant Notes | EduRev

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