# ABCD is a quadrilateral. Is AB + BC + CD + DA > AC + BD?

**Solution:**

In this question, it is asked to check Is AB + BC + CD + DA > AC + BD or not.

This question is based on the property 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 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.

**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

ABCD is a quadrilateral. Is AB + BC + CD + DA > AC + BD?

Hence AB + BC + CD + DA > AC + BD is true.