Question

'O' is any point inside a rectangle ABCD.Prove that OBsquare+ODsquare=OAsquare+OCsquare

Answer 1

Given ABCD is a rectangle so AD || BC, AB || CD and AD = BC , AB = CD
To prove OB² + OD² = OA² + OC²

Construction-- Draw a line passing through O and intersecting AD and BC at E And F respectively ..

Now In triangle DOE
ANGLE E = ANGLE A = 90°. { Corresponding angles }
therefore, OD²= ED²+OE² EQ ----1
Similarly in triangle BOF
OB² = BF²+OF² EQ-------2

On adding eq 1 and 2 we get,

OD²+OB² = ED²+OE²+BF²+OF² EQ -----3

Now In triangle AOE and triangle COF
Angle F = 90°
Angle E = 90° ( LINEAR PAIR)
Therefore
OA² = AE²+OE² EQ-----4
AND
OC² = OF²+FC² EQ -----5

On adding EQ 4 and 5 we get
OA²+OC² = AE²+OE²+OF²+FC² ----EQ 6

Since ABFE and EDCD are ||GM
Because AD || BC = AF || BF , ED || FC AND EF || AB)
NOW ,In EQ 6

OA²+OC² = BF²+OE²+ED²+OF² -EQ-------7
( BECAUSE AE = BF and ED = FC. Since opposites sides of a ||gm are equal... )

FROM EQ 3 AND 7 WE GET .....

OB²+OD² = OA²+OC²

PROVED.....

HOPE IT WILL BE HELPFUL.....