First slide

Properties of triangles

Question

If the angles of a triangle are 30° and 45° and the included side is $(\sqrt{3} + 1)$ ems, then the area of the triangle, is

Moderate

A

$\frac{1}{\sqrt{3} - 1}$
B

$\sqrt{3} + 1$
C

$\frac{1}{\sqrt{3} + 1}$
D

none of these

Solution

Using sine formula, we have
$\begin{array}{l} \frac{\sqrt{3} + 1}{\sin 105^{\circ}} = \frac{b}{\sin 30^{\circ}} = \frac{c}{\sin 45^{\circ}} \\ \Rightarrow 2 \sqrt{2} = 2 b = \sqrt{2} c \Rightarrow b = \sqrt{2}, c = 2 \end{array}$
$∴$ Area of $Δ A B C = \frac{1}{2} b c \sin A$
$\Rightarrow Area of △ A B C = \frac{1}{2} \times (2 \sqrt{2} \sin 105^{\circ}) = \frac{\sqrt{3} + 1}{2} = \frac{1}{\sqrt{3} - 1}$

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