Finding Length of Diagonals of Rhombus

[TrueStar]

Homework Statement

A rhombus has sides of length 100 centimeters and the angle at one of the vertices is 70 degrees. Approximate the lengths of the diagonals.

Homework Equations

Law of cosines

The Attempt at a Solution

I just want to make sure I have correct assumptions about a rhombus. If one vertices angle is 70 degrees, that means the opposite is as well. Since all four angles should equal 360 degrees, this means that the other remaining angles at 110 degrees each.

This is basically solving for a side of two triangle, I have two side lengths and an angle. My answers came out to about 114.7 centimeters and 163.8 centimeters.

Thanks Again :)

 

[LCKurtz]

Correct! You're welcome even though you didn't really have a question.

 

[Architeuthis Dux]

Hello! You are correct in your assumptions about the angles of a rhombus. However, it is important to note that the diagonals of a rhombus are not equal in length, unlike the sides. Therefore, you will need to use the law of cosines to solve for the lengths of the diagonals. The law of cosines states that for a triangle with sides a, b, and c and opposite angles A, B, and C, the following equation holds:

c^2 = a^2 + b^2 - 2ab*cos(C)

In this case, we can use this equation to solve for the lengths of the diagonals. Let's label the sides of the rhombus as follows:

a = 100 cm (length of one side)
b = 100 cm (length of another side)
C = 70 degrees (angle at one vertex)

We can then use the law of cosines to solve for the length of one diagonal, which we will call d1:

d1^2 = 100^2 + 100^2 - 2*100*100*cos(70)
d1^2 = 20000 - 20000*cos(70)
d1 = √(20000 - 20000*cos(70))
d1 = 114.7 cm

Similarly, we can solve for the length of the other diagonal, which we will call d2:

d2^2 = 100^2 + 100^2 - 2*100*100*cos(110)
d2^2 = 20000 - 20000*cos(110)
d2 = √(20000 - 20000*cos(110))
d2 = 163.8 cm

Therefore, the approximate lengths of the diagonals of the rhombus are 114.7 cm and 163.8 cm. I hope this helps!