Triangle is one of the most important and basic chapters. And it’s weightage in CAT is enough. As we all know, Triangle is the beginning of the Geometry section and many stucking with it Since IX standards. But When it to competition, it is not easy to retain geometry even though, we have studied it many times. So, here is some important formula for furnishing your basic concept.

The words Triangle has been made from Tri-angle. Where Three vertices connected to each other and creating Triangle.

Some Basic Points for Beginners-:

    • Every triangle has three vertices
    • The Sum of three angles is 180.
    • There are 6 types of triangle acute angle triangle, obtuse angle triangle, right triangle, Equilateral triangle, Isosceles triangle and Scalene triangle

Acute angle triangle :

A triangle with all sides less than 90º.

Obtuse angle triangle : 

An obtuse triangle has one side more than 90º.

Right angle triangle:

A right triangle with one side 90º. 

Here you can see Δ QRS = 90º

Equilateral triangle:

Triangle with all sides are equal and all angles measure the same angle 60º.

Isosceles triangle:

Triangle with two of its sides equal and consequently the angles opposite the equal sides are also equal.

Scalene triangle:

Triangles with none of the sides equal to any other sides.

Here is some fundamentals Rule:

  • The addition of any two sides must be greater than the then third sides.
    • a+b>c
    • b+c>a
    • c+a>b
  • The difference between any two sides must be lesser then third sides.
    • a-b<c
    • b-c<a
    • c-a<b
  • Side opposite the greatest angle will be greatest and the side opposite to the smallest angle the smallest.
  • The sin rule: a/sin A=b/sin B=C2R ( where R = circumradius )
  • The cosine rule: a² = b²+c²*2bc cosA
  • It is applicable to all sides and respective angles. triangle concept

In case of a right triangle, the formula reduces to 

a²=b²+c² Since cos 90º = 0

Exterior Angle

The Exterior angle is the sum of two interior angles not adjacent to it.

Exterior Angle I

As we have already discussed above the Sum of all Interior angle is 180. So here are A+B+ (Interior Angle C) = 180 Exterior angle C = A+B

Exterior Angle II

Let's try to understand... E+K+L( Interior Angle )= 180, Interior Angle "L" = 180-E+K, Interior Angle L= 180-112.6-17.2= 50.2, Exterior Angle L= 180-50.2= 129.9

 1. Area = 1/2 base * height or 1/2 bh.
Height= Perpendicular distance between the base and vertex opposite to it.
triangle concepttriangle concepttriangle concept

2. Area = √s(s-a)(s-b)(s-c)    (Heron’s formula)

Where = (a+b+c)/2

3. Inradius  ‘r’ & Semi perimeter ‘s’
Area = rs
Circum radius ‘R’
Area = abc/4R

 

 4. Area = 1/2 * product of two sides * sine of the included angle 
= 1/2 ac sin F
= 1/2 bc sin G
= 1/2 ab sin E

Congruency of Triangles

Two triangles are congruent if all sides of one triangle are equal to the corresponding sides of another. It follows that all the angles of one are equal to another corresponding angles of another. The congruency represent form (≅)

Type of Congruency

1. SAS (Side Angle Side) -: If One triangle’s side, angle and side equal to another triangle.

Here, AB = DE (Side)

∠B = ∠E (Angle)

BC = EF (Side)

ΔABC ≅ ΔDEF

triangle concept

2. ASA ( Angle Side Angle )-: If One triangle’s angle, side, and angle equal to another triangle.

Here, ∠A = ∠D (Angle)

AB = DE (Side)

∠B = ∠E (Angle)

ΔABC ≅ ΔDEF

trianlge concept

3. AAS ( Angle Angle Side )-: If two angles, side opposite to one angle are equal to the corresponding angles and the side of another triangle, the triangles are congruent.

Here, ∠B = ∠E (Angle)

∠C = ∠F (Angle)

AB = DE (Side)

ΔABC ≅ ΔDEF

triangle concept

4. SSS ( Side Side Side )-: If the three sides of one triangle are equal to three sides of another triangle. The triangle is congruent.

Here, AB = DE

BC = EF

AC = DF

ΔABC ≅  ΔDEF

triangle concept

5. SSA ( Side Side Angle )-: If two sides and one angle opposite to the greater side of a triangle is equal to the two sides and one angle opposite to the greater side of another triangle.

Here -: AB = DE

BC = EF

∠A = ∠D

ΔABC ≅ ΔDEF

triangle concept

6. RHS ( Right angle Hypotenuse Side ) -: If in two triangles the hypotenuse and the one side of one triangle are equal to the hypotenuse and one side of the other, triangle, then the two triangles are congruent.

Here, 

PQ ⊥ AB and PQ bisect AB. PQ intersects AB at C.

AP = BP

AQ = BQ

PQ = PQ

ΔPAQ ≅ ΔPBQ

Therefore,  ∠APQ = ∠BPQ

Let look towards ΔPAC and ΔPBC.

AP = BP

∠APC = ∠BPC

PC = PC

So, ΔPAC ≅ ΔPBC

Therefore, AC = BC

and ∠ACP = ∠BCP

Also, ∠ACP+∠BPC = 180º

2∠ACP = 180º

∠ACP = 90º

Trisngle concept

Similarity

The similarity of triangles is a special case where if either of the condition if the similarity is polygon holds, the other will hold automatically.

Types of similarity

  1. AAA similarity: if in two triangles, corresponding angles are equal, that is, the two triangles are equiangular then the triangles are similar.

Corollary (AA similarity): If two angles of one angle of another triangle then the two triangles are similar. The reason being, the third angle becomes equal automatically.

2. SSS similarity: if the corresponding sides of two triangles are proportional then they are similar.

For ΔABC to be similar to ΔPQR, AB/PQ = BC/QR = AC/PR must hold true.

3. SAS similarity: If in two triangles, one pair of corresponding sides are proportional and the included angles are equal then the two triangles are similar.

ΔABC∼ΔPQR

If AB/BC = PQ/QR and ∠B = ∠Q

The ratio of medians = Ratio of heights = Ratio of circumradius = Ratio of Inradius = Ratio of angle bisector

Properties of similar triangles

If the two triangles are similar, then for the proportional/ corresponding sides we have the following results.

triangle concept

  1. Ratio of sides = Ratio of heights (altitudes)

= Ratio of medians

= Ratio of angle bisector

= Ratio of Inradii

= Ratio of circumradii

2. Ratio of areas = Ratio of a square of the corresponding sides

i.e., If  ΔABC ∼ ΔPQR, then

= A(ΔABC)/A(ΔPQR) = (AB)²/(PQ)² = (BC)²/(QR)² = (AC)²/(PR)²

Two things are similar if they look similar. It may be smartphones, Car, father and son etc..This rule also apply to triangles.

EQUILATERAL TRIANGLE....

triangle concepttriangle concept

  1.                    ( sin 60 = √3/2 = h/side )

h = (a√3)/2

2. Area = 1/2 (base) * (height) = 1/2 *a*(a√3)/2 = √3 ⁄ 4 (a²)

3. R (circum radius) = (2h)/3 = a/√3

4. r (in radius) = h/3 = a/(2√3)

Properties

  1. The incenter and circumcenter lies at a pont that divides the height in th ration 2:1.
  2.  The circumradius us always twice the in radius. { R = 2r}
  3. Among all the triangles that can be formed with a given perimeter, the equilateral triangle will have the maximum area.
  4. An equilateral triangle in a circle will have the maximum area compared to other triangles inside the same circle

Isosceles Triangletriangle cocept

Area = (b/4) √4a²- b²,

In an isosceles triangle, the angles opposite to the equal sides are equal. 

Right Angled Triangle

Pythagoras Theorem In the case of a right-angle triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In the figure, for triangle ABC , a² =b² + c²

R(circumradius) = hypotenuse/2

Area = rs   (where r = in radius and s = (a+b+c)/2 where a,b and c are sides of the triangles)

⇒ 1/2 bc = r(a+b+c)/2

⇒ r = (bc)/(a+b+c)

triangle concept

In the triangle ABC,

ΔABC ∼ ΔDBA ∼ ΔDAC

(Note: A lot of questions are based on this figure.)

Further, we find the following identites:

  1. ΔABC ∼ ΔDBA

AB/BC = DB/BA

⇒ AB² = DB*BC

⇒ c² = pa

2. ΔABC ∼ ΔDAC

 AC/BC = DC/AC

⇒ AC² = DC * BC

⇒ b² = qa

3. ΔDBA ∼ ΔDAC

DA/DB = DC/DA

⇒ DA² = DB*DC

⇒ AD² = pq.

Important Pythagorean Triplets

|3,4,5|, |5,12,13|, |7,24,25|, |8,15,17|, |9,40,41|, |11,60,61|, |12,35,37|, |16,63,65|, |20,21,29|, |28,45,53|, |48,55,73|, |28,45,53|. These triplets are very important since a lot of questions are based on them.

Any triplet formed by either multiplying or dividing one of the basic triplets by any positive real number will be another Pythagorean triplet.

Thus, since 3,4,5 form a triplet so also will 6,8 and 10  as also 3.3, 4.4 and 5.5.

 

Two right triangles are similar if the hypotenuse and side of one is proportional to hypotenuse and side of another.

Here are Important Terms related to Triangles

Median a line joining from a vertex to the mid of opposite side is known as the median. in the figure, three medians are AD, BE and CF (D, E, and F are mid-point)

 

triangle concept

  • A midpoint divides a triangle into two equal parts.
  • A centroid is a point where three medians meet.
  • Centroid divides each median in the ratio 2:1.
  • Ac: cD = 2:1 = Bc :cF = Cc:cE

Formule 

Some of the squares of the other two sides = 2(median)² +2 * (1/2 the third side)²

⇒ (PQ)² + (PR)² = 2 (AD)² + 2(BC/2)²

2. Altitude height= a perpendicular drawn from any vertex to the opposite side is called the altitude. ( in the figure AD, BF and CE are the altitudes of the triangle)

    • all altitudes of a triangle meets at a point called the orthocenter of the triangle.
  • the angle made by any side at the orthocentre and Vertical angles make a supplementary pair. (i.e. they both add up to 180). in figure below
      • ∠A + ∠BOC = 180º = ∠C + ∠AOB

Triangle concept

3. perpendicular bisector a line that is perpendicular to a side and bisects it is the perpendicular bisector of the sides.

  • the point at which the perpendicular bisector of the sides meets is called circumcenter of the triangle.
  • does circumcenter is the center of the circle that circumscribes a triangle. there can be only one such circle.
  • The angle formed by any side at the circumcentre is two times the Vertical angle opposite to the side. this is the property of the circle whereby angles formed by an arc at the center are twice that of the angle formed by the same arc in the opposite arc.

 

4. Incenter

  • The line bisecting the interior angles of a triangle is the angle bisector of that triangle.
  • The angle bisector meets at a point called the incenter of the triangle.
  • The incenter is equidistant from all sides of the triangle.
  • From the incenter with a perpendicular draws to any of the sides as the radius, the circle can be drawn touching all the three sides. This is called the incircle of the triangle. The radius of the incircle is known as Inradius.
  • Triangle concept
  • The angle formed by any side at the incenter is always a right angle more than half the angle opposite to the side.
      • This can be understood by ∠QXR = 90+1/2∠P
  • If QY and RY are the angle bisector of the exterior angles at Q and R, then: 
          • ∠QYR = 90-1/2 ∠P

 

 

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