ABCD is a quadrilateral. Is AB + BC + CD + DA > AC + BD?
Solution:
This question is based on the triangle inequality theorem that the sum of lengths of two sides of a triangle is always greater than the third side.
Now visually identify that the quadrilateral ABCD. It is divided by diagonals AC and BD into four triangles.
Now, take each triangle separately and apply the above property and then add L.H.S and R.H.S of the equation formed.
In triangle ABC,
AB + BC > AC ……………(1)
In triangle ADC,
AD + CD > AC ………….. (2)
In triangle ADB,
AD + AB > DB --------------(3)
In triangle DCB,
DC+ CB > DB..................(4)
Adding equation (1), (2), (3) and (4) we get,
AB + BC + AD + CD + AD + AB + DC+ CB > AC + AC + DB + DB
AB + AB + BC + BC + CD + CD + AD + AD > 2AC + 2DB
2AB + 2BC + 2CD + 2AD > 2AC + 2DB
AB + BC + CD + AD > AC + DB
Hence, AB + BC + CD + DA > AC + BD is true
Useful Tip:
Whenever you encounter problems of this kind, it is best to think of the property based on the sum of lengths of any two sides of a triangle is always greater than the third side.
☛ Check: NCERT Solutions for Class 7 Maths Chapter 6
Video Solution:
ABCD is a quadrilateral. Is AB + BC + CD + DA > AC + BD?
NCERT Solutions for Class 7 Maths Chapter 6 Exercise 6.4 Question 4
Summary:
ABCD is a quadrilateral. Hence AB + BC + CD + DA > AC + BD is true.
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