Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev

Class 10 Mathematics by VP Classes

Class 10 : Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev

The document Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev is a part of the Class 10 Course Class 10 Mathematics by VP Classes.
All you need of Class 10 at this link: Class 10

Q.1. Let ΔABC ~ ΔDEF and their areas be, respectively, 64 cm2 and 121 cm2. If EF = 15.4 cm, find BC.
Sol. Since, ΔABC ~ ΔDEF The ratio of the areas of two similar triangles is equal to the ratio of the squares of the corresponding sides,
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
⇒  Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
⇒  Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev

Q2. 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.
Sol. ABCD is a trapezium with
AB || DC and AB = 2 CD
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
In ΔAOB and ΔCOD,
∠AOB = ∠COD   [V.O.A.]
∠1 = ∠2 [Alternate angles]
∴ ΔAOB ~ ΔCOD
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
⇒ ar ΔAOB: ar ΔCOD = 4 : 1

Q.3. In the figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that. Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
Sol. ABC and DBC are two triangles on the same base BC.
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
Construction: Draw AM ⊥ BC and DN ⊥ BC
Proof: In ΔAOM and ΔDON
∠AOM = ∠DON   [V.O.A.]
∠AMO = ∠DNO    [Each 90°]
∴  ΔAOM ~ ΔDON    [AA]
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
[Corresponding sides of similar triangles]
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev

Q.4. If the areas of two similar triangles are equal, prove that they are congruent.
Sol. 
Given: Consider ΔABC ~ ΔDEF
and ar ΔABC = ar ΔDEF
To Prove; ΔAABC ≌ ΔDEF
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
∴ ΔABC ≌ ΔDEF [By SSS congruency rule]

Q.5. D, E and F are respectively the mid-points of sides AB, BC and CA of D ABC. Find the ratio of the areas of ΔDEF and ΔABC.
Sol. Given: D, E and F are mid-points of sides BC, CA and AB respectively.
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
Proof: D and E are mid-points of sides BC and CA respectively
∴  Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
[Line segment joining the mid-points of two sides of triangle is parallel to the third side and half of it.]
Similarly
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
Hence ar ΔDEF : ar ΔABC = 1 : 4

Q.6. Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians.
Sol.
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
Given: ΔABC ~ ΔDEF, AP and DQ are medians
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
Proof: The ratio of the areas of two similar triangles is equal to the ratio of squares of two corresponding side
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev

Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
[Corresponding angles of similar triangle]
∴ ΔABP ~ ΔDEQ       [SAS]
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
From equation (i) and (ii)
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev

Q.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.
Sol. Let the side of the square ABCD be a
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
ΔPAD ~ ΔQAC    [AA similarity each angle = 60°]
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev

Q.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
Sol. Justification: Let AB = BC = CA = a
D is the mid point of BC
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
ΔABC ~ ΔBDE
[Given that they are equilateral triangles]
Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev

Q.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
Sol. Justification: Areas of two similar triangles are in the ratio of the squares of their corresponding sides.
∴  Ratio of areas of triangle Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev
or  16:81
Correct answer is (d)

Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!

Complete Syllabus of Class 10

Dynamic Test

Content Category

Related Searches

past year papers

,

Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev

,

mock tests for examination

,

practice quizzes

,

Semester Notes

,

Exam

,

ppt

,

video lectures

,

pdf

,

study material

,

Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev

,

Viva Questions

,

Ex 6.4 NCERT Solutions- Triangles Class 10 Notes | EduRev

,

Extra Questions

,

MCQs

,

Important questions

,

Free

,

Sample Paper

,

Objective type Questions

,

Previous Year Questions with Solutions

,

Summary

,

shortcuts and tricks

;