Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.
Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.
Let us consider, ABCD be a parallelogram. Now, draw perpendicular DE on extended side of AB, and draw a perpendicular AF meeting DC at point F.
By applying Pythagoras Theorem in ∆DEA, we get,
DE^{2} + EA^{2} = DA^{2} ……………….… (i)
By applying Pythagoras Theorem in ∆DEB, we get,
DE^{2} + EB^{2} = DB^{2}
DE^{2} + (EA + AB)^{2} = DB^{2}
(DE^{2} + EA^{2}) + AB^{2} + 2EA × AB = DB^{2}
DA^{2} + AB^{2} + 2EA × AB = DB^{2} ……………. (ii)
By applying Pythagoras Theorem in ∆ADF, we get,
AD^{2} = AF^{2} + FD^{2}
Again, applying Pythagoras theorem in ∆AFC, we get,
AC^{2} = AF^{2} + FC^{2} = AF^{2} + (DC − FD)^{2}
= AF^{2} + DC^{2} + FD^{2} − 2DC × FD
= (AF^{2} + FD^{2}) + DC^{2} − 2DC × FD. AC^{2}
AC^{2}= AD^{2} + DC^{2} − 2DC × FD ………………… (iii)
Since ABCD is a parallelogram,
AB = CD ………………….…(iv)
And BC = AD ………………. (v)
In ∆DEA and ∆ADF,
∠DEA = ∠AFD (Each 90°)
∠EAD = ∠ADF (EA  DF)
AD = AD (Common Angles)
∴ ∆EAD ≅ ∆FDA (AAS congruence criterion)
⇒ EA = DF ……………… (vi)
Adding equations (i) and (iii), we get,
DA^{2} + AB^{2} + 2EA × AB + AD^{2} + DC^{2} − 2DC × FD = DB^{2} + AC^{2}
DA^{2} + AB^{2} + AD^{2} + DC^{2} + 2EA × AB − 2DC × FD = DB^{2} + AC^{2}
From equation (iv) and (vi),
BC^{2} + AB^{2} + AD^{2} + DC^{2} + 2EA × AB − 2AB × EA = DB^{2} + AC^{2}
AB^{2} + BC^{2} + CD^{2} + DA^{2} = AC^{2} + BD^{2}

Nazima is fly fishing in a stream. The tip of her fishing rod is 1.8 m above the surface of the water and the fly at the end of the string rests on the water 3.6 m away and 2.4 m from a point directly under the tip of the rod.
3 years ago

In Figure, D is a point on side BC of ∆ ABC such that BD/CD = AB/AC. Prove that AD is the bisector of ∠BAC.
3 years ago

In Figure, two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that: (i) ∆ PAC ~ ∆ PDB (ii) PA.PB = PC.PD.
3 years ago

In Figure, two chords AB and CD intersect each other at the point P. Prove that : (i) ∆APC ~ ∆ DPB (ii) AP.PB = CP.DP
3 years ago

In Figure, AD is a median of a triangle ABC and AM ⊥ BC. Prove that : (i) AC^2 = AD^2 + BC.DM + 2 (BC/2)^2
3 years ago
 321 Forums
 27.3 K Topics
 53.8 K Posts
 2 Online
 12.4 K Members