In rhombus ABCD , prove that :

AB^2 + BC^2 + CD^2 + AD^2 = AD^2 + BD^2 

Asked by simranmangnani2282.9sdatl | 13th May, 2020, 02:52: PM

Expert Answer:

Question: In rhombus ABCD , prove that AB2 + BC2 + CD2 + AD2 = AC2 + BD2 
Solution:
In rhombus ABCD, AC and BD are diagonals.
Let the point of intersect of both the diagonals be O.
As we know that the digonals of a rhombus bisect each other at right angles, we have
angle A O B equals 90 degree comma space angle A O D equals 90 degree comma space angle B O C equals 90 degree comma space angle C O D equals 90 degree
A l s o comma space A O equals O C equals 1 half A C space a n d space B O equals O D equals 1 half B D
U sin g space P y t h a g o r a s space t h e o r e m space i n space triangle A O B comma
rightwards double arrow A B squared equals A O squared plus O B squared equals 1 fourth A C squared plus 1 fourth B D squared space... space left parenthesis i right parenthesis
U sin g space P y t h a g o r a s space t h e o r e m space i n space triangle A O D comma
rightwards double arrow A D squared equals A O squared plus O D squared equals 1 fourth A C squared plus 1 fourth B D squared space... space left parenthesis i i right parenthesis
U sin g space P y t h a g o r a s space t h e o r e m space i n space triangle B O C comma
rightwards double arrow B C squared equals B O squared plus O C squared equals 1 fourth B D squared plus 1 fourth A C squared space... space left parenthesis i i i right parenthesis
U sin g space P y t h a g o r a s space t h e o r e m space i n space triangle C O D comma
rightwards double arrow C D squared equals C O squared plus O D squared equals 1 fourth A C squared plus 1 fourth B D squared space... space left parenthesis i v right parenthesis
U sin g space left parenthesis i right parenthesis comma space left parenthesis i i right parenthesis comma space left parenthesis i i i right parenthesis space a n d space left parenthesis i v right parenthesis comma space w e space h a v e
A B squared plus B C squared plus C D squared plus D A squared equals 1 fourth A C squared plus 1 fourth B D squared plus 1 fourth B D squared plus 1 fourth A C squared plus 1 fourth A C squared plus 1 fourth B D squared plus 1 fourth A C squared plus 1 fourth B D squared
equals A C squared plus B D squared
H e n c e comma space A B squared plus B C squared plus C D squared plus D A squared equals A C squared plus B D squared

Answered by Renu Varma | 13th May, 2020, 03:33: PM