Advertisements
Advertisements
प्रश्न
Show that in a quadrilateral ABCD, AB + BC + CD + DA > AC + BD
Advertisements
उत्तर
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
संबंधित प्रश्न
ABC and DBC are two isosceles triangles on the same base BC (see the given figure). Show that ∠ABD = ∠ACD.
Show that the angles of an equilateral triangle are 60° each.
In Figure 10.24, AB = AC and ∠ACD =105°, find ∠BAC.
If the base of an isosceles triangle is produced on both sides, prove that the exterior angles so formed are equal to each other.
BD and CE are bisectors of ∠B and ∠C of an isosceles ΔABC with AB = AC. Prove that BD = CE.
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):
Angles opposite to equal sides of a triangle are equal
Which of the following statements are true (T) and which are false (F):
The two altitudes corresponding to two equal sides of a triangle need not be equal.
O is any point in the interior of ΔABC. Prove that
(i) AB + AC > OB + OC
(ii) AB + BC + CA > OA + QB + OC
(iii) OA + OB + OC >` 1/2`(AB + BC + CA)
Fill in the blank to make the following statement true.
The sum of three altitudes of a triangle is ..... than its perimeter.
Fill in the blank to make the following statement true.
If two sides of a triangle are unequal, then the larger side has .... angle opposite to it.
In a triangle ABC, if AB = AC and AB is produced to D such that BD = BC, find ∠ACD: ∠ADC.
In the given figure, if EC || AB, ∠ECD = 70° and ∠BDO = 20°, then ∠OBD is
In the given figure, for which value of x is l1 || l2?
In the given figure, the value of x is ______.
The base BC of triangle ABC is produced both ways and the measure of exterior angles formed are 94° and 126°. Then, ∠BAC =
In the given figure, if l1 || l2, the value of x is
