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