Question

If \[A = {30^\circ },c = 7\sqrt 3 \] and \[C = {90^\circ }\]in \[\Delta ABC\] then a =
A. \[7\sqrt 3 \]
B. \[\frac{{7\sqrt 3 }}{2}\]
C. \[\frac{7}{2}\]
D. None of these

Answer 1

Hint:
In order to find the answer to this question, you must use a right triangle. The angle at which A and c intersect is \[120\] degrees (A equals \[30\] degrees), so by adding these angles together, we get 180 degrees as the sum of both angles.
Formula used:
We can use sine properties to solve triangles:
\[\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}\]

Complete step-by-step solution:
Given that \[A = {30^\circ },c = 7\sqrt 3 \]and \[C = {90^\circ }\]in triangle \[ABC\].
To find the area of a triangle with this information we need to use basic trigonometry. To do so, we first need to identify the lengths of each side and then use Pythagorean theorem to determine the length of the hypotenuse. That includes the following:
\[A\] is the degree of a triangle's base
\[C\] is the length of a straight-line segment that goes through the vertex and ends at the other two vertices.
\[\Delta ABC\] is a right triangle
Lastly, you will use trigonometry to solve for \[a\].
Considering that \[C = {90^\circ }\]
\[a = \frac{{c\sin A}}{{\sin C}}\]
\[a = \frac{{7\sqrt 3 \sin {{30}^\circ }}}{{\sin {{90}^\circ }}}\]
\[a = \frac{{7\sqrt 3 }}{2}\]
So, Option B is correct.
Note:
Making sure that students comprehend how fundamental trigonometry functions is crucial. The altitude (\[A\]), as applied to right triangles, is a measurement of how far the triangle's peak angle rises above the opposing side. Here, \[A\] is at 30 degrees and c is at \[\frac{{7\sqrt 3 }}{2}\]. As a result, the angle between \[A\] and \[C\] is 90 degrees since the perpendicular bisector equally cuts across both positions.