Question

Acute angle of a rhombus whose side is a mean proportional between it's diagonals is ?

Answer 1

Let the diagonal are x, y in length so

side^2 = xy

Now we know that diagonal bisect at 90 degrees so

side^2 = (x/2)^2 + (y/2)^2

xy = x^2 + y^2/4

x^2 + y^2 - 4xy = 0

x/y + y/x = 4

y/x + 1/y/x = 4

Let y/x = t

t^2 -4t + 1 = 0

t = 4+- underoot 12/2 = 2 +- underoot 3

or y/x = 2 +- underoot 3

Now let the angle is θ

tan θ = y/x = 2- underoot 3  [We have to find acute angle only]

Now tan 15  = tan(45-30) = 1/-1/underoot 3 / 1+ 1/underoot 3 = 2 - underoot 3

So tan θ = tan 15

Hence θ = 15

Answer 2

Refer to the attached image.

Consider a rhombus ABCD,

Let the length of diagonal AC = 'x' and length of diagonal BD = 'y'.

Let the side of a rhombus be 'a' meters.

It is given that side is a mean proportional between it's diagonals.

Therefore, $a^2 = xy$

Since diagonals of rhombus are perpendicular bisectors.

Therefore, $OA = OC = \frac{x}{2} meter$

$OB = OD = \frac{y}{2} meter$

Consider triangle BOC,

By pythagoras theorem,

$BC^2 = OB^2 + OC^2$

$a^2 = (\frac{x}{2})^2 +(\frac{y}{2})^2$

$\frac{x^2+y^2}{4} = xy$

$x^2+y^2 = 4xy$

$1+(\frac{y}{x})^2= \frac{4y}{x}$

Let $\frac{y}{x} = t$

$1+(t)^2= 4t$

$t^2-4t+1 = 0$

$t = \frac{4\pm 2\sqrt{3}}{2}$

$t =2\pm \sqrt{3}$

Consider triangle DOC,

Let $\angle ODC = \Theta$

Consider $\tan \Theta = \frac{P}{B}$

$\tan \Theta = \frac{x}{2} \div \frac{y}{2}$

$\tan \theta = \frac{y}{x}$

Since, $\frac{y}{x} = t$

$\frac{y}{x}=2\pm \sqrt{3}$

$\frac{x}{y} = \frac{1}{2+\sqrt 3}$

Rationalizing, we get

$\frac{x}{y} = 2- \sqrt 3$

Therefore, $\tan \theta = 2 - \sqrt3$

Consider,

$\tan 15^{\circ}= \tan (45^\circ-30^\circ)$

$\tan 15^{\circ}= \frac{\tan 45^{\circ}-\tan 30^{\circ}}{1+\tan 45^{\circ}\tan 30^{\circ}}$

$\tan 15^{\circ}= \frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}$

$\tan 15^{\circ}= \frac{\sqrt{3}-1}{\sqrt{3}+1}$

On rationalizing, we get

$\tan 15^\circ = 2+ \sqrt3$

Hence, the acute angle of rhombus whose side is a mean proportional between it's diagonals is 15 degrees.