Advertisements
Advertisements
Question
Prove that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.
Advertisements
Solution
Diagonals of a parallelogram bisect each other, i.e., O is the midpoint of AC and BD.
In ∆ABD, point O is the midpoint of side BD.
BO = OD = `1/2`BD
∴ by Apollonius theorem,
AB2 + AD2 = 2AO2 + 2BO2 ...(1)
In ∆CBD, point O is the midpoint of side BD.
BO = OD = `1/2`BD
∴ by Apollonius theorem,
CB2 + CD2 = 2CO2 + 2BO2 ...(2)
Adding (1) and (2), we get,
AB2 + AD2 + CB2 + CD2 = 2AO2 + 2BO2 + 2CO2 + 2BO2
⇒ AB2 + AD2 + CB2 + CD2 = 2AO2 + 4BO2 + 2AO2 ...(∵ OC = OA)
⇒ AB2 + AD2 + CB2 + CD2 = 4AO2 + 4BO2
⇒ AB2 + AD2 + CB2 + CD2 = (2AO)2 + (2BO)2
⇒ AB2 + AD2 + CB2 + CD2 = (AC)2 + (BD)2
⇒ AB2 + AD2 + CB2 + CD2 = AC2 + BD2
∴ the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.
APPEARS IN
RELATED QUESTIONS
Adjacent sides of a parallelogram are 11 cm and 17 cm. If the length of one of its diagonal is 26 cm, find the length of the other.
Out of the following, which is the Pythagorean triplet?
Some question and their alternative answer are given.
In a right-angled triangle, if sum of the squares of the sides making right angle is 169 then what is the length of the hypotenuse?
Height and base of a right angled triangle are 24 cm and 18 cm find the length of its hypotenuse
Some question and their alternative answer are given. Select the correct alternative.
In ∆ABC, AB = \[6\sqrt{3}\] cm, AC = 12 cm, BC = 6 cm. Find measure of ∠A.
Find the height of an equilateral triangle having side 2a.
Find the length a diagonal of a rectangle having sides 11 cm and 60 cm.
In ∆RST, ∠S = 90°, ∠T = 30°, RT = 12 cm, then find RS and ST.
Sum of the squares of adjacent sides of a parallelogram is 130 sq.cm and length of one of its diagonals is 14 cm. Find the length of the other diagonal.
Seg PM is a median of ∆PQR. If PQ = 40, PR = 42 and PM = 29, find QR.
Seg AM is a median of ∆ABC. If AB = 22, AC = 34, BC = 24, find AM
If hypotenuse of a right angled triangle is 5 cm, find the radius of
the circle passing through all vertices of the triangle.
In ΔPQR, seg PM is the median. If PM = 9, PQ2 + PR2 = 290, Find QR.
In ΔABC, seg AP is a median. If BC = 18, AB2 + AC2 = 260, then find the length of AP.
Choose the correct alternative:
Out of given triplets, which is not a Pythagoras triplet?
"The diagonals bisect each other at right angles." In which of the following quadrilaterals is the given property observed?
Which of the following figure is formed by joining the mid-points of the adjacent sides of a square?
