The length of one diagonal of a
rhombus is a geometric mean
of the length of the other
diagonal and the length of the
side. Find angle measures of a
rhombus.
Answers
Answer:
Each side length = s unit & diagonals of a rhombus bisect at 90 deg angle.
BD^2 = d^2 = s^2 + s^2 -2*s*s* cosA (by cosine rule)
=> d^2 = 2s^2 - 2 cosA s^2 ……… (1)
In right triangle COD
OC = √{s^2 - (d/2)^2} = √(4s^2 - d^2)/2
=> AC = 2*OC = √(4s^2 - d^2) …… (2)
But AC is given = √(s*d)
By (2) , s*d = 4s^2 - d^2
=> d^2 = 4s^2 - sd …………. (3)
By equation (1) & (3)
4s^2 - sd = 2s^2 - 2cosA s^2
=> 2s^2 + 2cosAs^2 = sd
=> 2s^2 ( 1 + cosA ) = sd
=> 1 + cosA = d/2s
=> cosA = (d-2s)/2s ……….. (4)
Now, equation (3) is calculated further to solve for d in terms of s
d^2 + sd -4s^2 = 0 ( a quadratic equation in variable 'd' )
We get d = +,- (√17 -1) s / 2
Or, d = +,- 3.12s/2
Substitute this value in equation (4)
CosA = (+,- 3.12s/2 - 2s )/ 2s
=> cosA = ( 3.12s - 4s)/ 4s { -3.12 is ruled out as its cos value won't be defined}
=> -0.88/4 => cosA = - 0.22
=> <A = 102.71 degrees ✅
=> < B = 180–102.7 = 77.29 degrees✅