Friday, February 9th, 2018
Triangle is the one geometric figure that is the most basic and simple one but. Our first encounter with triangles starts when we are in Pre-school and taught about various shapes through interesting activities. And as soon as we commence stepping up the stairs of our schooling years from 1st till 12th standard, each year we learn something new about triangles. We began this triangular journey with learning about the basics and fundamental properties of the triangle in our 2nd– 3rd std. to concepts of congruency and similarity in our high school. And since triangle and its theory played such a centric and imperative role in our curriculum of mathematics whole through our school life, it’s importance cannot be undermined and therefore, has crucial importance in competitive exams such as CAT, Bank PO etc. Each year questions related to the application of concepts of triangle comes in the quantitative section of these exams. Thus, in this blog, triangle and its theory are discussed ranging from the basics with distinct types of triangles to the rules of congruency and similarity.
A triangle is a polygon with three edges and three vertices. It is formed by joining 3 non-collinear points in the 2-dimensional plane. And as the name itself signify the most elemental property of it i.e. the word triangle comes from joining Tri with angle where tri means three thus it has 3 angles that sum up to 180o where the 3 angles are the interior angles of the triangle given in the figure below.
And the sum of all the exterior angles is 360o.
The other basic properties of triangles are
Categorization of Triangles
Triangles can be divided into two types
|Based on length of side||Based on measure of angle|
In this type of triangle, the length of all the three sides is same and equivalent. Thus, the all the three angles are also equal i.e. 60o
AREA = √3/4*a2,
where a is the length of the side.
Acute angle triangle
|In an acute triangle, all the angles of the triangle are less 90o. An equilateral triangle is an acute triangle since all its angles are <90o. An isosceles and scalene triangle can also be an acute triangle.|
In this triangle, the length of two sides are equal and one is different.
Also, the angles corresponding to these sides are also equal.
AREA = ½ base x height
Obtuse Angle Triangle
|In an obtuse triangle, one angle measure greater than 90o. There cannot exist two obtuse angles in one triangle as the sum of all angles is 180o. Therefore, in the obtuse triangle one angle measure>90o and other two are acute. An equilateral triangle cannot be obtuse.|
|In a scalene triangle, all the sides measure different from each other and for the same reason the angles are also contrasting.||
Right Angle triangle
|A right triangle is the one in which one angle measures 90o and other two angles are acute and can be equal. The relation between the sides and angles of right triangle is the basis for trigonometry|
Now as we are done with describing diverse types of triangles we can move on to mensuration of triangles though, I have provided you some formulas and ways to measure the area of certain types of triangles but there’re many other methods also present in mathematics to estimate the area of triangles. In fact, triangles have the maximum number of ways to determine its area in comparison with other shapes. Following are the few ways that can be used to compute the area of triangle.
The most elementary formula for calculating the area of a triangle is ½ base x height but at times it’s difficult to compute the height of the triangle. No worries, you don’t have to find out the height the length of all the three sides will work. The Heron’s Formula evaluate area as follows
Let a, b, and c, be the length of three sides of triangle then,
Area = (s*(s-a)*(s-b)*(s-c))1/2,
Where, s = (a+b+c)/2
You can evaluate the area of a triangle using trigonometric function sine as follows.
Area = ½ bc x sinA = ½ ab x sinC = ½ ac x sin B
Determinants method use coordinate geometry to calculate the area of a triangle. Therefore,
when (x1, y1), (x2, y2), and (x3, y3) are vertices of triangle
There’re some other methods too to calculate the area of a triangle. The formulas for those are
Now, let’s move forward to understanding and knowing the difference between various line segments and centers such as Altitude, Medians etc. and Orthocenter, incenter etc. You all might have done questions and heard about them in your high school and many of you might be still confused with their terminology and basic differences among them. Given below is the table that lucidly describes each of its meaning.
An altitude is a line segment which passes through any vertex and forms the right angle with the edge opposite to this vertex.
Here, AD, CF, and BE are altitudes of a triangle.
The point of intersection of all the three altitudes in a triangle is known as orthocenter. The orthocenter may lie inside or outside of the triangle depending upon the type of triangle.
Here, o is the orthocenter.
A perpendicular bisector is a line segment which passes through any vertex of a triangle to the mid-point of the opposite side and makes right angle with it.
Here, AD, CE and BF are the perpendicular bisectors.
The intersection of all three perpendicular bisectors is known as circumcenter and it is also the center of the circle circumscribing the triangle.
Here, G is the circumcenter.
A median of a triangle is the line segment that joins any vertex of a triangle with the mid-point of the opposite side i.e. it divides the base to which it joins in two equal halves.
Here, QU, PT, and SR are three medians.
A triangle can have only three medians which intersect at the point known as the centroid. The centroid divides the length of the medians in 2:1 ratio.
Here, V is the centroid.
The angle bisector is the line segment that bisects the angle into two angles of equal measure of the vertex from which it is drawn.
It is the intersection of the three angle bisectors of the triangle. It is also the center of the incircle of the triangle.
Here, I is the incentre.
Congruency and Similarity
We all did many problems and proofs of theorems and rules of congruency and similarity. Some of these important rules that might assist you in the exam are given below in the table.
Two triangles are said to be congruent if they have the same size and shape i.e. all pairs of interior angles and corresponding sides measure the same.
Symbolically, we represent congruency between two triangles through
|Two triangles are said to be similar if every angle of one triangle has the same measure in the corresponding angle in other triangle and the corresponding sides in both the triangles are in the same ratio.|
|Rule 1: SAS||
Two triangles can be proved to be congruent if two sides of a triangle are equal to corresponding sides of another triangle and the angle between them is also of same measure.
Two triangles are similar if two pairs of sides in the two triangles are in same proportion with each other and corresponding angles in between the sides are also of equal measure.
|Rule 2: SSS||
Two triangles are congruent if all the three sides in one triangle are of the same measure as to corresponding sides in another triangle.
If the three sides of one triangle are in the same proportion with the corresponding three sides of the other then, they are said to be similar triangles.
|Rule 3: AAS/ AA||
If in two triangles, two corresponding angles are equal in measure and one non-included corresponding side is equal in length, then they are said to be congruent and this rule is thus known as Angle-Angle-side.
Two triangles are said to be similar if two pairs of angles are of same measure and this rule is known as Angle-Angle.
|Rule 4: ASA (Angle Side Angle)||
This rule only holds for congruency. It states that if two angles and included side in between them in one triangle are of equal measure to corresponding angles and side in the second one then, they are congruent.
|There’s no such rule applicable for similarity.|
Some Essential theorems:
We all know there’re innumerable theorems and postulates in mathematics and a few of those were taught us during our school time. And many of these theorems were related to triangles so some of them that hold substantial importance are mentioned below.
Angle Bisector Theorem: It states that the angle bisector of an angle in the triangle divides the opposite side internally into the ratio of the sides containing the angle. It means that, let there be a triangle ABC and AD be the angle bisector of this triangle and AD divides the side BC in m:n ratio then,
This theorem holds for both interior as well as exterior angles. The above-given figure is for the angle bisector bisecting the interior angle. For exterior angle,
Let there be a triangle ABC and AD be the angle bisector bisecting angle CAD of this triangle externally and AD divides the side BC in n:m ratio then,
The Pythagoras Theorem: It states that in a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides where hypotenuse is the side opposite to right side i.e. if there’s a triangle with the length of the sides containing the right angle be and b respectively and c, is the length of the hypotenuse then,
This theorem has paramount importance in the field of geometry with wide application and uses in many questions and theories.
Apollonius’s Theorem: It states that the sum of the square of any two sides of the triangle equals twice the sum of the square of half the third side and the median bisecting the third side. Let ABC be any triangle and AM, is the median of that triangle bisecting BC then,
Mid-point Theorem: It states that the line segment joining the mid-point of the two sides of the triangle is parallel to the third side is half of it. Let ATV be,
a triangle and R & S are the mid-points of the side AT and AV resp. then,
Basic Proportionality Theorem: This theorem states that if the line is drawn parallel to one side of the triangle to intersect the other two sides in distinct
points, the other two sides are divided in the same ratio. Let ABC be a triangle and DE be the line parallel to BC then,
This article has included all the details and concepts of triangle now you must learn their application and that comes with practice as “Practice is a means of inviting the perfection desired”.
All questions from CAT 2017 Quantitative Aptitude – Geometry
Quantitative Aptitude – Geometry – Triangles – Q1: Let P be an interior point of a right-angled isosceles triangle ABC with hypotenuse AB.
Quantitative Aptitude – Geometry – Triangles – Q2: Let ABC be a right-angled triangle with BC as the hypotenuse. Lengths of AB and AC are 15 km and 20 km, respectively.
Quantitative Aptitude – Geometry – Triangles – Q3: From a triangle ABC with sides of lengths 40 ft, 25 ft and 35 ft, a triangular portion GBC is cut off where G is the centroid of ABC.
Quantitative Aptitude – Geometry – Circles – Q1: ABCD is a quadrilateral inscribed in a circle with centre O. If ∠COD = 120 degrees and ∠BAC = 30 degrees
Quantitative Aptitude – Geometry – Circles – Q2: Let ABC be a right-angled isosceles triangle with hypotenuse BC. Let BQC be a semi-circle, away from A, with diameter BC.
Quantitative Aptitude – Geometry – Coordinate – Q1: The points (2, 5) and (6, 3) are two end points of a diagonal of a rectangle. If the other diagonal has the equation y = 3x + c, then c is
Quantitative Aptitude – Geometry – Coordinate – Q2: The shortest distance of the point (½, 1) from the curve y = |x -1| + |x + 1| is
Quantitative Aptitude – Geometry – Mensuration – Q1: The base of a vertical pillar with uniform cross section is a trapezium whose parallel sides are of lengths 10 cm and 20 cm
Quantitative Aptitude – Geometry – Mensuration – Q2: A ball of diameter 4 cm is kept on top of a hollow cylinder standing vertically.
Quantitative Aptitude – Geometry – Mensuration – Q3: A solid metallic cube is melted to form five solid cubes whose volumes are in the ratio 1 : 1 : 8: 27: 27.
Quantitative Aptitude – Geometry – Polygon – Q1: Let ABCDEF be a regular hexagon with each side of length 1 cm. The area (in sq cm) of a square with AC as one side is
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