Question

 The angles of the triangle are in ratio \[1:2:3\]. Find the difference between the smallest and largest angle (in degrees)

Answer 1

Hint: First we first assume that the angles of a triangle are \[x\], \[2x\] and \[3x\]. Then we will use the angle sum property of the triangle that the sum of the angles in a triangle is \[180^\circ \] to find the value of \[x\]. Then we will substitute the obtained value of \[x\] in the three angles of the triangle to find the difference between the smallest and largest angle.

Complete step by step answer:

We are given that the angles of a triangle are in the ratio \[1:2:3\].

Let us assume that the angles of a triangle are \[x\], \[2x\] and \[3x\].

We know the angle sum property of the triangle that the sum of the angles of the triangle is \[180^\circ \].

Then, we have

\[
   \Rightarrow x + 2x + 3x = 180^\circ \\
   \Rightarrow 6x = 180^\circ \\
 \]

Dividing the above equation by 6 on both sides, we get

\[
   \Rightarrow \dfrac{{6x}}{6} = \dfrac{{180^\circ }}{6} \\
   \Rightarrow x = 30^\circ \\
 \]

Substituting the above value of \[x\] in the three angles of triangle, we get

\[ \Rightarrow 1\left( {30^\circ } \right) = 30^\circ \]
\[ \Rightarrow 2\left( {30^\circ } \right) = 60^\circ \]
\[ \Rightarrow 3\left( {30^\circ } \right) = 90^\circ \]

Thus, the three angles are \[30^\circ \], \[60^\circ \] and \[90^\circ \].

Finding the difference of the largest angle \[30^\circ \] and the smallest angle \[90^\circ \], we get

\[ \Rightarrow 90^\circ - 30^\circ = 60^\circ \]

Thus, the required value is \[60^\circ \].

Note: In solving this question, we have multiplied the given ratio with some unknown variable. Then use the fact that the sum of the angle of a triangle is \[180^\circ \] to obtain a linear equation. Then this question is really simple to solve. Also, students should always remember that when all the angles are less than 90 degrees, then it is an acute angle and if one of the angles is 90 degrees then the triangle is right-angled.