Question

One angle of a rhombus measures 110°, and the shorter diagonal is 4 inches long. How long is the side of the rhombus? (Do not round until the final answer. Round angles to the nearest degree and side lengths to the nearest tenth of a unit.)

Answer 1

This is the concept of geometry, we are required to calculate for the length of the sides of the rhombus; we know that a rhombus is a compressed square, this implies that all the sides are equal;
If one of the angles is 110° the other angle will be:
180-110=70°
thus using the cosine rule we can find the side lengths as follows;
c^2=a^2+b^2-2ac Cos C
thus
let side a=b=x in
shorter diagonal=c=4 in
C=70°
substituting this into the formula we get:
4^2=x^2+x^2-2*x*x Cos 70
4^2=2x^2-2x^2(0.3420)
16=2x^2-2x^2(0.3420)
dividing through by 2 we get;
8=x^2-0.3420x^2
8=0.6580x^2
x^2=12.15843
getting the square root of both sides get:
x=sqrt(12.15843
x=3.4869)
x=3.5 (1 d.p)
the length of the sides is 3.5