Gujarat Board Solutions Class 10 Maths Chapter 6 Triangle Ex 6.4

Gujarat Board Solutions Class 10 Maths Chapter 6 Triangle Ex 6.4

Gujarat Board Textbook Solutions Class 10 Maths Chapter 6 Triangle 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:
We have
∆ABC ~ ∆DEF
ar ∆ABC = 64 cm2
ar ∆DEF = 14cm2
and EF = 15.4 cm
GSEB Solutions Class 10 Maths Chapter 6 Triangle Ex 6.4

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. (CBSE 2012)
Solution:
Trapezium ABCD, in which AB DC and diagonal AC and BD intersect at O.

In ∆AOB and ∆COD
∠AOB = ∠COD (Vertically opposite angle)
∠OAB = ∠OCD (Alternate Interior angles)
∴ ∆AOB ~ ∆COD (by AA similarity)
GSEB Solutions Class 10 Maths Chapter 6 Triangle Ex 6.4
∴ar ∆AOB: ar ∆COD = 4: 1

Question 3.
In Figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, arAABC AO
show that = GSEB Solutions Class 10 Maths Chapter 6 Triangle Ex 6.4 (CBSE 2005, 2012)

Solution:
∆ABC and IDBC are two triangles on the same base BC and diagonals AD and BC Intersect each other at O.
Let us draw AE ⊥ BC and DF ⊥ BC
In ∆AOE and ∆DOF
∠AOE = ∠DOF (Vertically opposite angles)
∠AEO = ∠DFO (each 900)
∆AOE ~ ∆DOF (by AA similarity)
AE/DF = AO/DO ………(1)
(Corresponding sides of two similar triangles are proportional)
GSEB Solutions Class 10 Maths Chapter 6 Triangle Ex 6.4
From equations (1) and (2)
GSEB Solutions Class 10 Maths Chapter 6 Triangle Ex 6.4

Question 4.
If the area of two similar triangles are equal, prove that they are congruent. (CBSE 2012)
Solution:
Given: ∆ABC and ∆DEF are two similar triangles such that, ar ∆ABC = ar ∆DEF

Question 5.
D, E and F are respectively the midpoints of sides AB, BC and CA of ∆ABC. Find the ratio of the area of ∆DEF and ∆ABC.
Solution:
D, E and F are respectively the midpoints of the sides of ∆ABC. Jour the points D, E and F to form ∆DEF.

∠ADF = ∠ABC (corresponding angles)
and ∆AFD = ∆ACE ……….(4)
(corresponding angles)
∴ ∆DEF ~ ∆ABC (by AA similarity)
GSEB Solutions Class 10 Maths Chapter 6 Triangle Ex 6.4
(The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides)
GSEB Solutions Class 10 Maths Chapter 6 Triangle Ex 6.4
Line drawn from mid points of two sides of a triangle is parallel to third and half of the third.
GSEB Solutions Class 10 Maths Chapter 6 Triangle Ex 6.4
ar ∆DEF : ar ∆ABC = 1 : 4

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:
Given: ∆ABC and ∆DEF such that
∆ABC ~ ∆DEF
AM and DN are their corresponding medians.


GSEB Solutions Class 10 Maths Chapter 6 Triangle Ex 6.4

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. (CBSE 2012)

Question 8.
ABC and BDE are two equilateral triangles such that D is the midpoint of BC. The ratio of the areas of triangles ABC and BDE
(a) 2 : 1
(b) 1 : 2
(c) 4 : 1
(d) 1 : 4
Solution:
∆ABC and ∆BDE are both equilateral triangles
∴ ∆ABC ~ ∆BDE (by AAA similarity)
GSEB Solutions Class 10 Maths Chapter 6 Triangle Ex 6.4
(Ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding sides)
GSEB Solutions Class 10 Maths Chapter 6 Triangle Ex 6.4
(D is the mid point of BC and AB = BC = CA)
∴ar ∆ABC : ar ∆BDE = 4 : 1
Thus correct option is (c).

Question 9.
Sides of two similar triangles are in the ratio 4: 9. The area of these triangles are in the ratio.
(a) 2 : 3
(b) 4 : 9
(c) 81 : 16
(d) 16 : 81
Solution:
Let ∆ABC and ∆DEF are similar then
GSEB Solutions Class 10 Maths Chapter 6 Triangle Ex 6.4
ar ∆ABC : ar ∆DEF = 16 : 81
Thus (d) is the correct answer.

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