Question

TRIGONOMETRY Find the length of the longer diagonal of this parallelogram round to the nearest tenth

Answer 1

Given the parallelogram ABCD

As shown: AB = 4 ft

m∠BAC = 30

m∠BDC = 104

We will find the length of the longer diagonal which will be AC

See the following figure:

The point of intersection of the diagonals = O

The opposite sides are parallel

AB || CD

m∠ABD = m∠BDC because the alternate angles are congruent

So, in the triangle AOB, the sum of the angles = 180

m∠AOB = 180 - (30+104) = 46

We will find the length of OA using the sine rule as follows:

`\begin{gathered} (OA)/(\sin104)=(AB)/(\sin 46) \\ \\ OA=AB\cdot(\sin104)/(\sin46)=4\cdot(\sin104)/(\sin46)\approx5.3955 \end{gathered}`

The diagonals bisect each other

So,

`AC=2\cdot OA=10.79`

The longer diagonal is AC

Rounding to the nearest tenth

So, the answer will be AC = 10.8 ft