Question

Sides of a triangle are \[2\;{\rm{cm}},\sqrt 6 \;{\rm{cm,}}\]and\[\left( {\sqrt 3 + 1} \right){\rm{cm}}\]. The smallest angle of the triangle is
A. \[30^\circ \]
B. \[45^\circ \]
C. \[60^\circ \]
D. \[75^\circ \]

Answer 1

Hint:
The triangle's lengths, widths, and heights can be used to calculate the sides of a triangle, although this solution isn't always accurate. In rare circumstances, one side may be wider or shorter than the other.
Use of fundamental geometric concepts like proportionality and likeness is crucial when resolving issues like these. You can use these ideas to work out which measures connect to which triangle sides and how they relate to one another. Once these relationships have been established, include them into your equation to allow you to solve for the required sides.
Complete step-by-step solution:
Given that \[a = 2,b = \sqrt 6 ,c = \sqrt 3 + 1\]
\[\cos A = \frac{{6 + 3 + 1 + 2\sqrt 3 - 4}}{{2\sqrt 6 (\sqrt 3 + 1)}}\]
Calculate the sum or difference
\[\cos A = \frac{{6 + 2\sqrt 3 }}{{2\sqrt 6 (\sqrt 3 + 1)}}\]
Factor out 2 from the expression
\[\cos A = \frac{{2\left( {3 + \sqrt 3 } \right)}}{{2\sqrt 6 \left( {\sqrt 3 + 1} \right)}}\]
Cancel out the common factor 2
\[\cos A = \frac{{\left( {3 + \sqrt 3 } \right)}}{{\sqrt 6 \left( {\sqrt 3 + 1} \right)}}\]
Rationalize the denominator
 \[\cos A = \frac{{\left( {3 + \sqrt 3 } \right)\sqrt 6 }}{{6\left( {\sqrt 3 + 1} \right)}}\]
Distribute \[\sqrt 6 \]through the parentheses
\[\cos A = \frac{{3 + \sqrt 6 \sqrt {18} }}{{\sqrt 6 \left( {\sqrt 3 + 1} \right)}}\]
Simplify the radical expression
\[\cos A = \frac{{3 + \sqrt 6 + 3 + \sqrt 2 }}{{\sqrt 6 \left( {\sqrt 3 + 1} \right)}}\]
Factor out 3 from the expression
\[\cos A = \frac{{3\left( {\sqrt 6 + \sqrt 2 } \right)}}{{\sqrt 6 \left( {\sqrt 3 + 1} \right)}}\]
Factor out \[\sqrt 2 \] from the expression
\[\cos A = \frac{{\sqrt 2 \left( {\sqrt 3 + 1} \right)}}{{2\left( {\sqrt 3 + 1} \right)}}\]
Cancel out the common factor \[\sqrt 3 + 1\]
\[\cos A = \frac{{\sqrt 2 }}{2}\]
Using the unit circle, find the angles for which the cosine equals \[\frac{{\sqrt 2 }}{2}\]
As a result, \[A = {45^\circ }\].
So, Option B is correct.
Note:
Students must consider the length and width of each side in order to solve this issue. Since \[\left( {\sqrt 3 + 1} \right)\]cm is shorter than \[2cm\], the shorter of the two sides must be the other. Furthermore, it must be the smaller side because it is \[\sqrt 6 cm\] wider than both sides.
As students try to determine the angle that is smaller than \[45^\circ \], they may mistakenly assume \[72^\circ \] since the triangle is an equal right triangle with two separate sides.