Question

The largest angle of a triangle, if the ratios of its angles are \[2:3:4\], is:
A. 40
B. 80
C. 90
D. 120

Answer 1

Hint: We will first consider the given angles as \[2x,3x\] and \[4x\]. As we know that the triangle has three angles, so the sum of all three angles is equal to 180. From this, we will find the value of \[x\]. Next, we will substitute the value of \[x\] in the angles and find the largest angle of the triangle.

Complete step by step answer:

We are given that the ratios of the angle of the triangle are \[2:3:4\].
The aim is to find the largest angle of the triangle.
We will first consider the angles as \[2x,3x\] and \[4x\].
As the above angles are the angles of the triangle, so, the sum of all three angles is equal to 180.
Thus, we get,
\[ \Rightarrow 2x + 3x + 4x = 180\]
Now, we will further simplify the above expression to determine the value of \[x\].
Hence, we have,
\[
   \Rightarrow 9x = 180 \\
   \Rightarrow x = 20^\circ \\
 \]
Hence, we get the value of \[x\] as \[20^\circ \]
Next, we will substitute the value of \[x\] in the angles \[2x,3x\] and \[4x\] to find which angle is the largest angle of the triangle.
Hence, we get,
\[ \Rightarrow 2x = 2\left( {20^\circ } \right) = 40^\circ \]
\[ \Rightarrow 3x = 3\left( {20^\circ } \right) = 60^\circ \]
\[ \Rightarrow 4x = 4\left( {20^\circ } \right) = 80^\circ \]
Hence, from the above three angles, we can conclude that the angle \[80^\circ \] is the largest angle of the triangle.
Thus, option B is correct.

Note: We have used the angle sum property of the triangle in which the sum of all the angles of the triangle is 180. After finding the values of angles, we can easily determine the largest angle from all the angles of the triangle. The conversion of ratios into the angle is possible after multiplying the same variable to the ratio given. After evaluating the value of \[x\], it is necessary to do the substitution to find the largest angle.