Question

In the triangle shown below the value of \[y\] is twice the value of \[x\] and the value of \[z\] is three times the value of \[y\] then calculate the value of \[x\]

(a) 20
(b) 24
(c) 30
(d) 36

Answer 1

Hint: We solve this problem by using the sum of angles of a triangle. We use the standard condition that the sum of all angles of a triangle is equal to \[{{180}^{\circ }}\] that is for the given triangle we have
\[\Rightarrow \angle A+\angle B+\angle C={{180}^{\circ }}\]
Then we use the given condition that the relations between the angles find the required value.

Complete step-by-step solution
We are given that the angles as
\[\begin{align}
  & \Rightarrow \angle A=y \\
 & \Rightarrow \angle B=z \\
 & \Rightarrow \angle C=x \\
\end{align}\]
We are given that the value of \[y\] is twice the value of \[x\]
By converting the above statement into mathematical equation we get
\[\Rightarrow y=2x\]
We are also given that the value of \[z\] is three times the value of \[y\]
Now, by converting the above statement into mathematical equation we get
\[\Rightarrow z=3y\]
Now, by substituting the value of \[y\] in the above equation we get
\[\begin{align}
  & \Rightarrow z=3\left( 2x \right) \\
 & \Rightarrow z=6x \\
\end{align}\]
We know that the standard condition that the sum of all angles of a triangle is equal to \[{{180}^{\circ }}\] that is for the given triangle we have
\[\Rightarrow \angle A+\angle B+\angle C={{180}^{\circ }}\]
Now, by substituting the values of angles in above equation we get
\[\Rightarrow x+y+z=180\]
Now, by substituting the value of \[y,z\] in terms of \[x\] in above equation we get
\[\begin{align}
  & \Rightarrow x+2x+6x=180 \\
 & \Rightarrow 9x=180 \\
 & \Rightarrow x=20 \\
\end{align}\]
Therefore the value of \[x\] is 20.
So, option (a) is the correct answer.

Note: Students may do mistake in taking the value of \[z\]
We are given that the \[z\] is three times the value of \[y\] then we get
\[\Rightarrow z=3y\]
Here, the value of \[y\] depends on \[x\] as \[y=2x\]
But students miss this dependency of \[y\] on \[x\] and take the value of \[z\] as
\[\Rightarrow z=3x\]
This is not correct because all the angles given are mutually dependent on each other. So, we need to take the dependency of all angles and convert all angles in terms of one angle as we did here, that is we converted all the angles in terms of \[x\] which is required.
Here, we may be asked to find the remaining angles also which will be the continuous question.
So, by substituting the value of \[x\] in the formula \[y=2x\] we get
\[\begin{align}
  & \Rightarrow y=2\times 20 \\
 & \Rightarrow y=40 \\
\end{align}\]
Similarly, by substituting the value of \[x\] in the formula \[z=6x\] we get
\[\begin{align}
  & \Rightarrow z=6\times 20 \\
 & \Rightarrow z=120 \\
\end{align}\]
Therefore the remaining angles are 40 and 120.