Answer
b = 2
c = 2$\sqrt 3$ and
d = $\sqrt 6$
Work Step by Step
In given figure 28, 'a' is the longest side, 'b' is the shortest side and 'c' is the side opposite $60^{\circ}$ of a 30°–60°–90° triangle-
The longest side of a 30°–60°–90° triangle is twice the shortest side and the side opposite the 60° angle is $\sqrt 3$ times the shortest side.
Given that
Longest side, a = 4
Therefore
Shortest side, b = $\frac{longest side}{2}$ = $\frac{4}{2}$ = 2
Side opposite 60° i.e. c = $\sqrt 3\times $ shortest side = $\sqrt 3 \times$2 = 2$\sqrt 3$
Side opposite 60° i.e. c = 2$\sqrt 3$
Now 'c' is the hypotenuse and d is one of the shorter sides of a 45°–45°–90° triangle. Hence-
c = $d\times\sqrt 2$ = $d\sqrt 2$
Therefore d =$ \frac{c}{\sqrt 2} $ = $ \frac{2\sqrt 3}{\sqrt 2} $
d = $ \frac{\sqrt 2\times\sqrt 2\times\sqrt 3}{\sqrt 2} $ ( writing 2 as $\sqrt 2\times\sqrt 2$)
d= $\sqrt 2\times\sqrt 3$ = $\sqrt 6$
