Exercise 6.4
Is it possible to have a triangle with the following sides?
(i) 2 cm , 3 cm , 5 cm
(ii) 3 cm , 6 cm , 7 cm
(iii) 6 cm , 3 cm , 2 cm
Sol :
In a triangle, the sum of the lengths of either two sides is always greater than the third side
(i) Given that , the sides of the triangle are 2 cm , 3 cm , 5 cm
It can be observed that ,
2 + 3 = 5 cm
However, 5 cm = 5 cm
Hence, this triangle is not possible.
(ii) Given that, the sides of the triangle are 3 cm, 6 cm, 7 cm.
Here, 3 + 6 = 9 cm > 7 cm
6 + 7 = 13 cm > 3 cm
3 + 7 = 10 cm > 6 cm
Hence, this triangle is possible.
(iii) Given that, the sides of the triangle are 6 cm, 3 cm, 2 cm.
Here, 6 + 3 = 9 cm > 2 cm
However, 3 + 2 = 5 cm < 6 cm
Hence, this triangle is not possible.
Q 2, Ex 6.4 - The Triangle and its Properties - Chapter 6 - Maths Class 7th - NCERT
QUESTION 2
Take any point O in the interior of a triangle PQR . ls
(i) OP + OQ > PQ ?
(ii) OQ + OR > QR ?
(iii) OR + OP > RP ?
Sol :
If O is a point in the interior of a given triangle, then three triangles △OPQ, △OQR, and △ORP can be constructed . In a triangle , the sum of the lengths of either two sides is always greater than the third side
(i) Yes, as △OPQ is a triangle with sides OR , OQ , and PQ.
OP + OQ > PQ
(ii) yes , as △OQR is a triangle with sides OR , OQ and QR
OQ + OR > QR
(iii) yes , as △ORP is a triangle with sides OR , OP and PR
OR + OP > PR
Q 3, Ex 6.4 - The Triangle and its Properties - Chapter 6 - Maths Class 7th - NCERT
QUESTION 3
AM is a median of a triangle ABC.
Is AB + BC + CA > 2 AM ?
(Consider the sides of triangles △ABM and △AMC)
Sol :
In a triangle, the sum of the lengths of either two sides is always greater than the third side.
In △ABM,
AB + BM > AM ....(i)
Similarly, in △ACM ,
AC + CM > AM ...(ii)
Adding equation (i) and (ii) ,
AB + BM + MC + AC > AM + AM
AB + BC + AC > 2 AM
Yes, the given expression is true.
Q 4, Ex 6.4 - The Triangle and its Properties - Chapter 6 - Maths Class 7th - NCERT
QUESTION 4
ABCD is quadrilateral.
Is AB + BC + CD + DA > AC + BD ?
Sol :
In a triangle, the sum of the lengths of either two sides is always greater than the third side.
Considering △ABC.
AB + BC > CA ...(i)
In △BCD ,
BC + CD > DB ...(ii)
In △CDA
CD + DA > AC
In △DAB,
DA + AB > DB ..(iii)
Adding equations (i) , (ii) , (iii) and (iv) , we obtain
AB + BC +BC +CD +CD +DA + DA +AB > AC + BD + AC +BD
2AB + 2BC + 2CD + 2DA > 2AC + 2BD
2(AB + BC + CD + DA ) > 2(AC + BD)
(AB + BC + CD + DA ) > (AC + BD)
Yes , the given expression is true
Q 5, Ex 6.4 - The Triangle and its Properties - Chapter 6 - Maths Class 7th - NCERT
QUESTION 5
ABCD is quadrilateral.
Is AB + BC + CD + DA < 2(AC + BD) ?
Sol :
In a triangle , the sum of the lengths of either two sides is always greater than the third side.
Considering △OAB ,
OA + OB > AB ..(1)
In △OBC
OB + OC > BC ..(ii)
In △OCD ,
OC + OD > CD ..(iii)
In △ODA ,
OD + OA > DA ..(iv)
Adding equations (i) , (ii) , (iii) and (iv) , we obtain
OA + OB + OB + OC + OC + OD + OA > AB + BC + CD + DA
2OA + 2OB + 2OC + 2OD > AB + BC + CD + DA
2OA + 2OC + 2OB + 2OD > AB + BC + CD + DA
2(OA + OC) + 2(OB + OD) > AB + BC + CD + DA
2(AC) + 2(BD) > AB + BC + CD + DA
2(AC + BD) > AB + BC + CD + DA
Yes , the given expression is true
Q 6, Ex 6.4 - The Triangle and its Properties - Chapter 6 - Maths Class 7th - NCERT
QUESTION 6
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