06. Triangles

Exercise 6.1

Question 1:

Fill in the blanks using correct word given in the brackets:−
(i) All circles are __________. (congruent, similar)
(ii) All squares are __________. (similar, congruent)
(iii) All __________ triangles are similar. (isosceles, equilateral)
(iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are __________ and (b) their corresponding sides are __________. (equal, proportional)

Answer:

(i) Similar
(ii) Similar
(iii) Equilateral
(iv) (a) Equal
(b) Proportional

Question 2:

Give two different examples of pair of
(i) Similar figures
(ii)Non-similar figures

Answer:

(i) Two equilateral triangles with sides 1 cm and 2 cm

Two squares with sides 1 cm and 2 cm

(ii) Trapezium and square

Triangle and parallelogram
  

Question 3:

State whether the following quadrilaterals are similar or not:
 
Answer:
Quadrilateral PQRS and ABCD are not similar as their corresponding sides are proportional, i.e. 1:2, but their corresponding angles are not equal.


Exercise 6.2

Question 1:
 

In figure.6.17. (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).
(i)

(ii)
 
Answer:
(i)

Let EC = x cm
It is given that DE || BC.
By using basic proportionality theorem, we obtain

(ii)

Let AD = x cm
It is given that DE || BC.
By using basic proportionality theorem, we obtain
 

Question 2:

E and F are points on the sides PQ and PR respectively of a ΔPQR. For each of the following cases, state whether EF || QR.
(i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm
(ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm
(iii)PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.63 cm

Answer:

(i)

Given that, PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm, FR = 2.4 cm

(ii)

PE = 4 cm, QE = 4.5 cm, PF = 8 cm, RF = 9 cm

(iii)

PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm, PF = 0.36 cm


 
Question 3:

In the following figure, if LM || CB and LN || CD, prove that

 
Answer:


In the given figure, LM || CB
By using basic proportionality theorem, we obtain
 

Question 4:

In the following figure, DE || AC and DF || AE. Prove that

 
Answer:

In ΔABC, DE || AC


 

Question 5:
 

In the following figure, DE || OQ and DF || OR, show that EF || QR.
 
Answer:


In Δ POQ, DE || OQ



  
Question 6:

In the following figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR.
 

Answer:

In Δ POQ, AB || PQ



 

 Question 7:
Using Basic proportionality theorem, prove that a line drawn through the mid-points of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).
Answer:

Consider the given figure in which PQ is a line segment drawn through the mid-point P of line AB, such that


Or, Q is the mid-point of AC. 

Question 8:
Using Converse of basic proportionality theorem, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).

Answer:


Consider the given figure in which PQ is a line segment joining the mid-points P and Q of line AB and AC respectively.
i.e., AP = PB and AQ = QC
It can be observed that

Hence, by using basic proportionality theorem, we obtain

 

Question 9:
ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that  
Answer:

Draw a line EF through point O, such that
In ΔADC,
By using basic proportionality theorem, we obtain

In ΔABD,
So, by using basic proportionality theorem, we obtain

From equations (1) and (2), we obtain
  

Question 10:
The diagonals of a quadrilateral ABCD intersect each other at the point O such that Show that ABCD is a trapezium. 

Answer:

Let us consider the following figure for the given question.

Draw a line OE || AB

In ΔABD, OE || AB
By using basic proportionality theorem, we obtain

However, it is given that

⇒ EO || DC [By the converse of basic proportionality theorem]
⇒ AB || OE || DC
⇒ AB || CD
∴ ABCD is a trapezium.