7th Class Mathematics The Triangle and its Properties Basic Concepts of Triangles

     Basic Concepts of Triangles               

 

The general shape of a triangle is shown below:                

               

The vertices of a triangle are denoted by the Capital letters of English alphabets.                

In the above figure \[\Delta ABC,\]the sides are AB, BC and CA.                

 

      Altitude                

A Perpendicular drawn from a vertex to the opposite side is called the altitude of the triangle and denoted as h.  

 

      Some Basic Facts Related to Triangle

• In any triangle, sum of any two sides is always greater than the 3rd side. i.e \[b+c>a\]or \[a+c>b\]or \[c+b>a.\]
• The sum of all interior angles of a triangle is \[180{}^\circ \]
• The exterior angle at any vertex of the triangle is equal to the sum of other two opposite angles.                

                 

Proof: In \[\Delta ABC,a+b+c=180{}^\circ \]                

but,\[c+x=180{}^\circ \]   (Linear pair) \[c={{180}^{0}}-x\]                

Putting "c" in the above equation, we get                

\[a+b+180{}^\circ -x=180{}^\circ \Rightarrow a+b=x\]                   

Therefore,\[~\Delta ACD=\Delta A+\Delta B\]  

 

      Types of Triangle                

Classification based on angles

• If one angle of a triangle is right angle then it is called right angled triangle.

Note that the other two angles are acute.

• From figure, \[\angle B=90{}^\circ \]and \[\angle A+\angle C=90{}^\circ .\]So, it is right-angled triangle

• If all the angles of triangle are less than 90° then it is called acute-angled triangle.                

In the figure \[\angle A,\angle B\]and \[\angle C\]are acute angles.

• If one angle of a triangle is more than 90° then it is called obtuse angled triangle. The other two angles are acute.  

 

      Equilateral Triangle

A triangle in which all sides are equal is known as equilateral triangle.

 

      Isosceles Triangle                

A triangle in which any two sides are equal is said to be and isosceles triangle.                

The angles opposite to the equal sides are equal.                

In the triangle given below, sides AB and AC are equal as well as \[\angle B\]and \[\angle C\]are also equal.                

 

 

      Scalene Triangle                

A triangle in which all side are unequal is said to be scalene triangle.                

In scalene triangle all angles are different.                

               

In the above given triangle all the sides of triangle denoted by a, b and c are unequal and angles x, y and z are also unequal.                

 

       Pythagoras Theorem                

In a right angled triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides, mathematically from figure,

        

 

               

 

In right angled triangle AB = 3 unit, BC = 4 unit then AC equal to                

               

(a) 3 unit                                             

(b) 4 unit                

(c) 5 unit                                             

(d) 6 unit                

(e) None of these                

 

Answer: (c)                

Explanation                

We know that from Pythagoras theorem                

\[A{{B}^{2}}+B{{C}^{2}}=A{{C}^{2}}{\Rightarrow }A{{C}^{2}}=\]\[{{3}^{2}}+{{4}^{2}}{\Rightarrow }9+16=25={{5}^{2}}{\Rightarrow }AC=5unit.\]                                

 

Converse of Pythagoras Theorem                  

If the square of one side of a triangle is equal to the sum of the square of other two sides then the triangle is right angled triangle, where right angle is opposite to the greatest side.                

If in a triangle \[PQR,\text{ }P{{Q}^{2}}=P{{R}^{2}}+Q{{R}^{2}}\]then the angle opposite to PQ is right angle.  

 

      Pythagorean Triplets                

Three integers p, q and r (such that p > q > r) are said to be a Pythagorean triplet, if \[{{p}^{2}}={{q}^{2}}+{{r}^{2}}\]  

 

 

 

 

In the adjoining figure, BC is produced to D and CA is produced to E, and \[\angle DCA=108{}^\circ \]and \[\angle EAB=124{}^\circ \]then the value of \[x\] is:                

               

(a) 48°                                                  

(b) 52°                

(c) 76°                                                  

(d) 128°                

(e) None of these                                

 

Answer: (b)                

Explanation                                                     

\[\angle ACB=180{}^\circ -\angle ACD\]                

\[\Rightarrow {{180}^{0}}-{{108}^{0}}={{72}^{0}}\Rightarrow \angle BAC={{180}^{0}}-EAB\]                

\[\Rightarrow 180{}^\circ -124{}^\circ =56{}^\circ .\]From \[\angle ABC,\]we have                                     

\[\Rightarrow x+(\angle ACB+\angle BAC)=\]\[{{180}^{0}}\Rightarrow x+({{72}^{0}}+{{56}^{0}})={{180}^{0}}\]                

\[\Rightarrow x=180{}^\circ -128{}^\circ =52{}^\circ \]                  

 

 

If the angles of a triangle are in the ratio 1:1:2 then which one of the following statements is incorrect? 

(a) Triangle is right angled triangle                

(b) The angles of the triangles are \[90{}^\circ ,\text{ }45{}^\circ \] and\[~45{}^\circ ~\]                

(c) The angles of the triangles are \[90{}^\circ ,\text{ }45{}^\circ \] and \[45{}^\circ .\]Triangle is right angled isosceles triangle.                        

(d) The angles of the triangles are \[90{}^\circ ,\text{ }45{}^\circ \] and \[45{}^\circ .\] And it is scalene.                

(e) None of these                                

 

Answer: (d)                

Explanation                

The angles of triangle are \[x,\text{ }x\]and \[2x\] therefore, from angle sum property of triangle we get \[x+x+2x=180{}^\circ \] or, \[4x=180{}^\circ \]

or \[x=45{}^\circ ,\]the other angles of the triangle are \[90{}^\circ ,\text{ }45{}^\circ \] and \[45{}^\circ .\]                

Here two angles are equal. Therefore, the given triangle is isosceles triangle.                

 

 

If the bisector of an angle of a triangle is also the median of the triangle then the triangle in which this condition is not possible?                

(a) Equilateral                                                   

(b) Isosceles                

(c) Equiangular                                                 

(d) Scalene                

(e) None of these                                

 

Answer: (c)  

 

 

If CE is parallel to DB in the given figure then value of \[x\] will be____.                

(a) \[45{}^\circ \]                                              

(b) \[75{}^\circ \]                

(c)\[30{}^\circ \]                                              

(d) \[85{}^\circ \]                

(e) None of these                                

 

Answer: (d)                

                 

               

\[\angle BDC=75{}^\circ -40{}^\circ =35{}^\circ \]                

\[DB\left| \text{ } \right|CE\]                

\[\therefore 35{}^\circ +\left( 600+x{}^\circ  \right)=180{}^\circ \]                           

\[\Rightarrow x=85{}^\circ \]                                  

 

 

In the given figure P is the point on side BC. Which one of the following is correct?                

               

(a) \[(AB+BC+CA)<2AP\]             

(b) \[(AB+BC+CA)>2AP\]                

(c)\[(AB+BC+CA)<AP\] 

(d) \[~(AB+BC+CA)>AP\]                

(e) None of these                

 

Answer: (b)    

 

 

 

• A triangle is a polygon with three sides.
• Triangle is classified on the basis of angles and sides.
• According to angle there are three types of triangle

(i) acute angled triangle

(ii) right angled triangle

(iii) obtuse angled triangle.

• According to sides triangles are of also three types

(i) equilateral triangle

(ii) isosceles triangle

(iii) scalene triangle.

• Median is a line segment which joins the midpoint of a side and its opposite vertex.
• Incentre is the point of intersection of angle bisectors of a triangle.
• The point of intersection of altitude of a triangle is known as orthocenter.
• The point of intersection of perpendicular bisectors of the sides of triangle is known as circumventer.
• The point of intersection medians of a triangle is known as centroid. Hypotenuse is the longest side of a right angled triangle.      

 

 

 

 

• Pascal's triangle is an arithmetical triangle made up of staggered rows of numbers.