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
ΔABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see the given figure). Show that ∠BCD is a right angle.
In Figure AB = AC and ∠ACD =105°, find ∠BAC.
AB is a line seg P and Q are points on opposite sides of AB such that each of them is equidistant from the points A and B (See Fig. 10.26). Show that the line PQ is perpendicular bisector of AB.
Two lines AB and CD intersect at O such that BC is equal and parallel to AD. Prove that the lines AB and CD bisect at O.
ABC is a triangle in which BE and CF are, respectively, the perpendiculars to the sides AC and AB. If BE = CF, prove that ΔABC is isosceles
Fill the blank in the following so that the following statement is true.
Angle opposite to equal sides of a triangle are .....
Fill the blank in the following so that the following statement is true.
In a ΔABC if ∠A = ∠C , then AB = ......
Fill the blank in the following so that the following statement is true.
If altitudes CE and BF of a triangle ABC are equal, then AB = ....
In ΔABC, side AB is produced to D so that BD = BC. If ∠B = 60° and ∠A = 70°, prove that: (i) AD > CD (ii) AD > AC
Which of the following statements are true (T) and which are false (F)?
Sum of any two sides of a triangle is greater than the third side.
Which of the following statements are true (T) and which are false (F)?
Difference of any two sides of a triangle is equal to the third side.
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 the given figure, for which value of x is l1 || l2?
In the given figure, AB and CD are parallel lines and transversal EF intersects them at Pand Q respectively. If ∠APR = 25°, ∠RQC = 30° and ∠CQF = 65°, then
If the bisectors of the acute angles of a right triangle meet at O, then the angle at Obetween the two bisectors 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.
Find all the angles of an equilateral triangle.
In a triangle ABC, D is the mid-point of side AC such that BD = `1/2` AC. Show that ∠ABC is a right angle.
