Question

A point inside a rectangle ABCD is joined to the vertex prove that the sum of area of pair of opposite angles so formed is equal to the sum of the area of other pair of triangle

Answer 1

we have a rectangle ABCD

Point O is inside the rectangle where all the Vertices meet

In Triangles ACD and BCD

Angles ACD= BDC = 90° (Angle of between two sides of a rectangle is  90  degree)

AC = BD (Opposite sides are equal)CD=CD (Common)Triangles ACD is congruent to BCD (By Side Angle Side)

Hence ,Area of ACD = Area of BCD

ACD - OCD = BCD - OCD  (By Subtracting OCD from both the triangles)

We get,

AOC = BOD (proved)....(i)

In Triangles BAC and ACD

AB = CD (opposite sides are equal in a rectangle)

Angles BAC = DCA = 90°

Thus, Triangles BAC is congruent to ACD (by Side Angle Side)Areas of BAC = ACD 

BAC - AOC = ACD - AOC (Subtracting AOC from both triangles)

AOB = OCD (proved)...(ii)

Since,

Triangles BAC = ACD = BCD (Proved earlier)

AOC = BOD = AOB = OCD....(iii)

AOC + BOD = AOB + OCD

LHS  

AOC + AOC (Since AOC = BOD) 

=2AOC

From (iii)

AOC = AOB2AOC = 2AOB 

RHS

AOB + AOB (SInce OCD =AOB)=2AOB