Advertisements
Advertisements
प्रश्न
Show that in a quadrilateral ABCD, AB + BC + CD + DA < 2(BD + AC)
Advertisements
उत्तर
Given: ABCD is a quadrilateral.
To show: AB + BC + CD + DA < 2(BD + AC)
Construction: Join diagonals AC and BD.
Proof: In ΔOAB, OA + OB > AB ...(i) [Sum of two sides of a triangle is greater than the third side]
In ΔOBC, OB + OC > BC ...(ii) [Sum of two sides of a triangle is greater than the third side]
In ΔOCD, OC + OD > CD ...(iii) [Sum of two sides of a triangle is greater than the third side]
In ΔODA, OD + OA > DA ...(iv) [Sum of two sides of a triangle is greater than the third side]
On adding equations (i), (ii), (iii) and (iv), we get
2[(OA + OB + OC + OD] > AB + BC + CD + DA
⇒ 2[(OA + OC) + (OB + OD)] > AB + BC + CD + DA
⇒ 2(AC + BD) > AB + BC + CD + DA ...[∵ OA + OC = AC and OB + OD = BD]
⇒ AB + BC + CD + DA < 2(BD + AC)
APPEARS IN
संबंधित प्रश्न
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
ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see the given figure). Show that
- ΔABE ≅ ΔACF
- AB = AC, i.e., ABC is an isosceles triangle.
Find the measure of each exterior angle of an equilateral triangle.
Determine the measure of each of the equal angles of a right-angled isosceles triangle.
In a ΔPQR, if PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP
respectively. Prove that LN = MN.
Which of the following statements are true (T) and which are false (F):
Angles opposite to equal sides of a triangle are equal
Fill the blank in the following so that the following statement is true.
Angle opposite to equal sides of a triangle are .....
Which of the following statements are true (T) and which are false (F)?
Sum of the three sides of a triangle is less than the sum of its three altitudes.
Which of the following statements are true (T) and which are false (F)?
Sum of any two sides of a triangle is greater than twice the median drawn to the third side.
Fill in the blank to make the following statement true.
If two angles of a triangle are unequal, then the smaller angle has the........ side opposite to it.
In the given figure, the sides BC, CA and AB of a Δ ABC have been produced to D, E and F respectively. If ∠ACD = 105° and ∠EAF = 45°, find all the angles of the Δ ABC.
ABC is a triangle. The bisector of the exterior angle at B and the bisector of ∠C intersect each other at D. Prove that ∠D = \[\frac{1}{2}\] ∠A.
If the angles of a triangle are in the ratio 2 : 1 : 3, then find the measure of smallest angle.
Line segments AB and CD intersect at O such that AC || DB. If ∠CAB = 45° and ∠CDB = 55°, then ∠BOD =
In the given figure, if AB ⊥ BC. then x =
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 =
Which of the following correctly describes the given triangle?
Show that in a quadrilateral ABCD, AB + BC + CD + DA > AC + BD
