Answer:
3.5 in (nearest tenth)
Step-by-step explanation:
Properties of a rhombus:
- Quadrilateral (four sides & four interior angles)
- Parallelogram (opposite sides are parallel)
- All sides are equal in length
- Opposite angles are equal in measure
- Diagonals bisect each other at right angles
- Interior angles sum to 360°
- Adjacent angles are supplementary (sum to 180°)
- Diagonals bisect interior angles
Therefore, a rhombus is made up of 4 congruent right triangles.
** see attached diagram **
To find the side length of the rhombus, we need to calculate the hypotenuse of the right triangle.
As the shorter diagonal is 4 in, the base of the right triangle is 2 in
The angles that measure 110° are the angles by the shorter diagonal. Therefore, the base angle of the right triangle is 55°
Using cos trig ratio:
[tex]\sf \cos(\theta)=\dfrac{A}{H}[/tex]
where:
- [tex]\theta[/tex] is the angle
- A is the side adjacent the angle
- H is the hypotenuse
Given:
- [tex]\theta[/tex] = 55°
- A = 2
- H = x
[tex]\implies \sf \cos(55^{\circ})=\dfrac{2}{x}[/tex]
[tex]\implies \sf x=\dfrac{2}{\cos(55^{\circ})}[/tex]
[tex]\implies \sf x=3.486893591...[/tex]
Therefore, the side of the rhombus is 3.5 in (nearest tenth)
