Chapter 6 Triangles Ex 6.4

Question 1.
Let ∆ABC ~ ∆DEF and their areas be, respectively, 64 cm2 and 121 cm2. If EF = 15.4 cm, find BC.
Solution:
Since, ∆ABC ~ ∆DEF
The ratio of the areas of two similar triangles is equal to the ratio of the squares of the corresponding sides.
NCERT Solutions for Class 10 Maths Chapter 6 Triangles Ex 6.4 1

Question 2.
Diagonals of a trapezium ABCD with AB || DC intersect each other at the point O. If AB = 2 CD, find the ratio of the areas of triangles AOB and COD.
Solution:
ABCD is a trapezium with AB || DC and AB = 2 CD
NCERT Solutions for Class 10 Maths Chapter 6 Triangles Ex 6.4 2

Question 3.
In the given figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that
NCERT Solutions for Class 10 Maths Chapter 6 Triangles Ex 6.4 3
NCERT Solutions for Class 10 Maths Chapter 6 Triangles Ex 6.4 4
Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles Ex 6.4 5

Question 4.
If the areas of two similar triangles are equal, prove that they are congruent.
Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles Ex 6.4 6
NCERT Solutions for Class 10 Maths Chapter 6 Triangles Ex 6.4 7

Question 5.
D, E and F are respectively the mid-points of sides AB, BC and CA of ∆ABC. Find the ratio of the areas of ∆DEF and ∆ABC.
Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles Ex 6.4 8

Question 6.
Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.
Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles Ex 6.4 9
NCERT Solutions for Class 10 Maths Chapter 6 Triangles Ex 6.4 10

Question 7.
Prove that the area of an equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonals.
Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles Ex 6.4 11

Question 8.
ABC and BDE are two equilateral triangles such that D is the mid-point of BC. Ratio of the areas of triangles ABC and BDE is
(a) 2 :1
(b) 1:2
(c) 4 :1
(d) 1:4
Solution:
NCERT Solutions for Class 10 Maths Chapter 6 Triangles Ex 6.4 12

Question 9.
Sides of two similar triangles are in the ratio 4:9. Areas of these triangles are in the ratio
(a) 2:3
(b) 4:9
(c) 81:16
(d) 16:81
Solution:
Justification: Areas of two similar triangles are in the ratio of the squares of their corresponding sides.
NCERT Solutions for Class 10 Maths Chapter 6 Triangles Ex 6.4 13