Advertisements
Advertisements
Question
Show that in a quadrilateral ABCD, AB + BC + CD + DA > AC + BD
Advertisements
Solution
Given in the question, a quadrilateral ABCD.
To proof that AB + BC + CD + DA > AC + BD.
Proof: In triangle ABC,
AB + BC > AC ...(i) [Sum of the lengths of any two sides of a triangle must be greater than the third side]
In triangle BCD,
BC + CD > BD ...(ii) [Sum of the lengths of any two sides of a triangle must be greater than the third side]
In triangle CDA,
CD + DA > AC ...(iii) [Sum of the lengths of any two sides of a triangle must be greater than the third side]
Similarly, in triangle DAB,
AD + AB > BD ...(iv) [Sum of the lengths of any two sides of a triangle must be greater than the third side]
Now, adding equation (i), (ii), (iii) and (iv), we get
AB + BC + BC + CD + CD + DA + AD + AB > AC + BD + AC + BD
2AB + 2BC + 2CD > 2AC + 2BD
2(AB + BC + CD + DA) > 2(AC + BD)
AB + BC + CD + DA > AC + BD
Hence proved.
APPEARS IN
RELATED QUESTIONS
In an isosceles triangle ABC, with AB = AC, the bisectors of ∠B and ∠C intersect each other at O. Join A to O. Show that:
- OB = OC
- AO bisects ∠A
Show that the angles of an equilateral triangle are 60° each.
In a ΔABC, if ∠A=l20° and AB = AC. Find ∠B and ∠C.
If the base of an isosceles triangle is produced on both sides, prove that the exterior angles so formed are equal to each other.
PQR is a triangle in which PQ = PR and S is any point on the side PQ. Through S, a line is drawn parallel to QR and intersecting PR at T. Prove that PS = PT.
ABC is a triangle in which ∠B = 2 ∠C. D is a point on BC such that AD bisects ∠BAC and AB = CD.
Prove that ∠BAC = 72°.
Which of the following statements are true (T) and which are false (F):
Sides opposite to equal angles of a triangle may be unequal
Which of the following statements are true (T) and which are false (F):
The measure of each angle of an equilateral triangle is 60°
Fill the blank in the following so that the following statement is true.
In an isosceles triangle ABC with AB = AC, if BD and CE are its altitudes, then BD is …… CE.
Which of the following statements are true (T) and which are false (F)?
If two angles of a triangle are unequal, then the greater angle has the larger side opposite to it.
Fill in the blank to make the following statement true.
Difference of any two sides of a triangle is........ than the third side.
In the given figure, if EC || AB, ∠ECD = 70° and ∠BDO = 20°, then ∠OBD is
In the given figure, what is y in terms of x?
The base BC of triangle ABC is produced both ways and the measure of exterior angles formed are 94° and 126°. Then, ∠BAC =
In ∆PQR, ∠R = ∠P and QR = 4 cm and PR = 5 cm. Then the length of PQ is ______.
If ∆PQR ≅ ∆EDF, then is it true to say that PR = EF? Give reason for your answer
M is a point on side BC of a triangle ABC such that AM is the bisector of ∠BAC. Is it true to say that perimeter of the triangle is greater than 2 AM? Give reason for your answer.
Is it possible to construct a triangle with lengths of its sides as 9 cm, 7 cm and 17 cm? Give reason for your answer.
In the following figure, D and E are points on side BC of a ∆ABC such that BD = CE and AD = AE. Show that ∆ABD ≅ ∆ACE.
