5.
Figure 2
Diagram NOT accurately drawn
Figure 2 shows the plan view of a frame for a flat roof.
The shape of the frame consists of triangle \(A B D\) joined to triangle \(B C D\).
Given that
- \(B D = x \mathrm {~m}\)
- \(C D = ( 1 + x ) \mathrm { m }\)
- \(B C = 5 \mathrm {~m}\)
- angle \(B C D = \theta ^ { \circ }\)
- show that \(\cos \theta ^ { \circ } = \frac { 13 + x } { 5 + 5 x }\)
Given also that
- \(x = 2 \sqrt { 3 }\)
- angle \(B A C = 30 ^ { \circ }\)
- \(A D C\) is a straight line
- find the area of triangle \(A B C\), giving your answer, in \(\mathrm { m } ^ { 2 }\), to one decimal place.