Question

One angle of a rhombus measures 110, and the shorter diagonal is 4 inches long. How long is the side of the rhombus?

Answer 1

side of rhombus: 2.44 inch

Use the sine rule:

`\sf (sin(A))/(a) = (sin(B))/(b)`

============= Let the side be "b"

`\rightarrow \sf (sin(110))/(4) = (sin(35))/(b)`

`\rightarrow \sf b = (sin(35)*4)/(sin(110))`

`\rightarrow \sf b = 2.441549178`

`\rightarrow \sf b =2.44 \ in`

Answer 2

3.5 in (nearest tenth)

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:

`\sf \cos(\theta)=(A)/(H)`

where:

• 
  `\theta` is the angle
• A is the side adjacent the angle
• H is the hypotenuse

Given:

• 
  `\theta` = 55°
• A = 2
• H = x

`\implies \sf \cos(55^(\circ))=(2)/(x)`

`\implies \sf x=(2)/(\cos(55^(\circ)))`

`\implies \sf x=3.486893591...`

Therefore, the side of the rhombus is 3.5 in (nearest tenth)