Question

In the given figure (not drawn to scale), \[{\text{EFA}}\] is a right-angled triangle with \[\angle {\text{EFA = 90}}^\circ \] and \[{\text{FGB}}\] is an equilateral triangle, find the value of \[y - 2x\].

Answer 1

Hint: Here, in the question, we have been given a figure having multiple triangles and measurement of some of the angles is also given either in the numerical form or in the variable form. First we have to find the values of both the variables and then the value of expression given. We will use the sum property of angles of the triangle to get the desired result.

Complete step by step solution:
Given \[{\text{EFA}}\] is a right-angled triangle with \[\angle {\text{EFA = 90}}^\circ \]
\[{\text{FGB}}\] is an equilateral triangle
Considering \[\vartriangle {\text{FGB}}\], We know that,
\[\angle {\text{FGB + }}\angle {\text{FBG + }}\angle {\text{BFG = 180}}^\circ \] [Sum property of angles of a triangle]
Also, we know that all the angles of an equilateral triangle are equal. Therefore,
\[
  \angle {\text{FGB + }}\angle {\text{FGB + }}\angle {\text{FGB = 180}}^\circ \\
   \Rightarrow 3\angle {\text{FGB}} = 180^\circ \\
   \Rightarrow \angle {\text{FGB}} = 60^\circ \;
 \]
Now, consider \[\vartriangle {\text{CFG}}\]
\[\angle {\text{CFG + }}\angle {\text{CGF + }}\angle {\text{GCF = 180}}^\circ \] [Sum property of angles of a triangle]
\[
   \Rightarrow x + 60^\circ + 92^\circ = 180^\circ \\
   \Rightarrow x = 180^\circ - 152^\circ \\
   \Rightarrow x = 28^\circ \;
 \]
Now, we have \[\angle {\text{BFG = }}60^\circ \]
\[ \Rightarrow \angle {\text{BFC + }}\angle {\text{GFC = 60}}^\circ \] \[\left[ {\because \angle {\text{BFG = }}\angle {\text{BFC + }}\angle {\text{GFC}}} \right]\]
\[
   \Rightarrow \angle {\text{BFC + }}28^\circ = 60^\circ \\
   \Rightarrow \angle {\text{BFC = 32}}^\circ \;
 \]
Also, we have \[\angle {\text{EFA = 90}}^\circ \]
\[ \Rightarrow \angle {\text{EFB + }}\angle {\text{BFC = 90}}^\circ \] \[\left[ {\because \angle {\text{EFA = }}\angle {\text{EFB + }}\angle {\text{BFC}}} \right]\]
\[
   \Rightarrow y + {\text{32}}^\circ = 90^\circ \\
   \Rightarrow y{\text{ = 58}}^\circ \;
 \]
To find: \[y - 2x\]
\[y - 2x = 58^\circ - 2\left( {28^\circ } \right)\]
Simplifying it, we get,
\[
  y - 2x = 58^\circ - 56^\circ \\
   \Rightarrow y - 2x = 2^\circ \;
 \]

Note: Whenever we face such types of questions, it is important that we understand and remember the basic properties of different types of triangles. Like we have used, in the given question, all the angles of an equilateral triangle are equal and also the sum property of angles of triangle. Given any type of triangle, this property always holds true. Sum of angles of any triangle will always be equal to \[180^\circ \].