Question

A rhombus has side lengths of 25. What could be the lengths of the diagonals? *

Answer 1

Minimum value: 0, Maximum value: 50

Explanation:

A rhombus is a quadrilateral with side of same length but different angle pairs. Let consider the following rhombus (See attachment). It is known that sum of all angles in quadrilateral is equal to 360°. The length of each diagonal is, respectively:

`d_(1) = 2\cdot l\cdot \sin 0.5\alpha`

`d_(2) = 2\cdot l \cdot \sin (90-0.5\alpha)`

`d_(2) = 2\cdot l \cdot (\sin 90^(\circ)\cdot \cos 0.5\alpha - \cos 90 \cdot \sin 0.5\alpha)`

`d_(2) = 2\cdot l \cdot \cos 0.5\alpha`

Each expression shows that value of a diagonal changes at the expense of the other one inasmuch angle changes. The minimum and maximum values of any of the diagonal occur when trigonometric functions are equal to 0 and 1, respectively.

Minimum value:
`d_(min) = 0`, Maximum value:
`d_(max) = 2\cdot l = 2 \cdot (25) = 50`

Answer 2

0 < x < 50

Explanation:

We start out with a square (which IS a rhombus for all sides are equal in

length. That's when the diagonals are equal in length, which, by the

Pythagorean theorem equal to

5*√2

C^2=a^2+b^2

=25^2+25^2

Factorise

c^2=25^2 × 2

c=square root of 25^2 × 2

That is

c=√25^2 × 2

=5×√2

As we decrease the angle on the bottom left and increase the angle on

the bottom right, the green diagonal increases to 25+25 or 50, but never gets to 50. The red diagonal shrinks to 0 but never gets to 0.

the lengths of a diagonal can only be in the open interval from 0 to 50. In interval notation that is (0,50) or 0 < x < 50.