Question

What is the area of the triangle shown in the figure?

A. $18\sqrt 3$
B. $9 + 9\sqrt 3$
C. $9 + 18\sqrt 3$
D. $18 + 18\sqrt 3$

Answer 1

Hint : Area is the quantity that expresses the extent of a two-dimensional shape. Let $\angle C$ as x. The sum of all the angles of the triangle is equal to 180 degrees. Using $\sin \theta$ to find the area of the triangle. The area of a triangle is a measurement of the area covered by the triangle.

Complete step-by-step answer :
As we know, the sum of all angles inside the triangle will be equal to 180.
In the triangle ABC,
$\angle A + \angle B + \angle C = {180^ \circ }$
As given in triangle,
$\angle A = {45^ \circ }$
$\angle B = {105^ \circ }$
And $\angle C = x$
Calculate the value of x:
So,
$\angle C = 180 - (105 + 45)$
$\angle C = {30^ \circ }$
Using Sine rule,
$\dfrac{{\operatorname{Sin} A}}{a} = \dfrac{{\operatorname{Sin} C}}{c}$
Keeping value of c = 6
$\dfrac{1}{{a\sqrt 2 }} = \dfrac{1}{{12}}$
Solving it for a,
$a = 6\sqrt 2$
The area of the triangle is given by $\dfrac{{ac\operatorname{Sin} B}}{2}$
Keeping value in this from above we get,
$\dfrac{{6\sqrt 2 \times 6 \times \sin {{105}^ \circ }}}{2}$
$\sin {105^ \circ }$ can also be written as
$\sin {105^ \circ } = \sin {75^ \circ } = \sin (30 + 45) = \sin {30^ \circ }\cos {45^ \circ } + \sin {45^ \circ }\cos {30^ \circ }$
Used the formula of $\sin (A + B)$
$\operatorname{Sin} {105^ \circ } = \dfrac{1}{{2\sqrt 2 }} + \dfrac{{\sqrt 3 }}{{2\sqrt 2 }}$
So I will solve it further. We get,
$\dfrac{{1 + \sqrt 3 }}{{2\sqrt 2 }}$
Keeping it in area formula,
Thus the area is $\dfrac{{36\sqrt 2 \times (1 + \sqrt 3 )}}{{4\sqrt 2 }}$ $= 9 + 9\sqrt 3$
Hence, the area of the triangle is $9 + 9\sqrt 3$. So, the correct option is option (B).

Note : The area of a triangle is a measurement of the area covered by the triangle. The area of a triangle is determined by two formulas i.e. the base multiplies by the height of a triangle divided by 2 and second is Heron’s formula. Heron's formula is a method for calculating the area of a triangle when the lengths of all three sides of the triangle are given. These angles are formed by two sides of the triangle, which meets at a common point, known as the vertex. The sum of all three interior angles is equal to 180 degrees.