Question

The angles of a triangle are in the ratio \[{\text{3:4:5}}\]. Find the smallest angle.

Answer 1

Hint: In this question, we have given the ratio of the angle of the triangle. Let us assume the ratio be x. After that we know that the sum of interior angle of a triangle is always \[{\text{18}}{{\text{0}}^{\text{o}}}\]. So add all the angles of a triangle and equate it with \[{\text{18}}{{\text{0}}^{\text{o}}}\] on solving that we will get the required answer.

Complete step by step solution: We have given that the ratio of angles of a triangle is \[{\text{3:4:5}}\]
Let the angle of a triangle are \[\angle {\text{A}}\], \[\angle {\text{B}}\] and \[\angle {\text{C}}\]
Then, \[\angle {\text{A:}}\angle {\text{B:}}\angle {\text{C}}\]\[{\text{ = 3:4:5}}\]
Let the ratio be x.
Then we get,
\[\angle {\text{A = 3x}}\], \[\angle {\text{B = 4x}}\] and \[\angle {\text{C = 5x}}\]
As we know that the sum of interior angle of a triangle is always \[{\text{18}}{{\text{0}}^{\text{o}}}\]
Therefore, we have
\[\angle {\text{A + }}\angle {\text{B + }}\angle {\text{C = 18}}{{\text{0}}^{\text{o}}}\]
\[ \Rightarrow {\text{3x + 4x + 5x = 18}}{{\text{0}}^{\text{o}}}\]
\[ \Rightarrow {\text{12x = 18}}{{\text{0}}^{\text{o}}}\]
\[ \Rightarrow {\text{x = }}\dfrac{{{\text{18}}{{\text{0}}^{\text{o}}}}}{{{\text{12}}}}\]
\[ \Rightarrow {\text{x = 1}}{{\text{5}}^{\text{o}}}\]
Consider,
\[\angle {\text{A = 3x}}\]
Put \[{\text{x = 1}}{{\text{5}}^{\text{o}}}\] so, we get
\[\angle {\text{A = 3}} \times {\text{1}}{{\text{5}}^{\text{o}}}\]
\[ \Rightarrow \angle {\text{A = 4}}{{\text{5}}^{\text{o}}}\]
Consider,
\[\angle {\text{B = 4x}}\]
Put \[{\text{x = 1}}{{\text{5}}^{\text{o}}}\]so, we get
\[\angle {\text{B = 4}} \times {\text{1}}{{\text{5}}^{\text{o}}}\]
\[ \Rightarrow \angle {\text{B = 6}}{{\text{0}}^{\text{o}}}\]
Similarly we do it for \[\angle {\text{C = 5x}}\]
Put \[{\text{x = 1}}{{\text{5}}^{\text{o}}}\] so, we get
\[\angle {\text{C = 5} \times {1}}{{\text{5}}^{\text{o}}}\]
\[ \Rightarrow \angle {\text{C = 7}}{{\text{5}}^{\text{o}}}\]
So, we have \[\angle {\text{A = 4}}{{\text{5}}^{\text{o}}}{\text{,}}\:\angle {\text{B = 6}}{{\text{0}}^{\text{o}}}{\text{,}}\:\angle {\text{C = 7}}{{\text{5}}^{\text{o}}}\]

So we can observe that, \[\angle {\text{A = 4}}{{\text{5}}^{\text{o}}}\] is the smallest angle of a triangle.

Note: A triangle has three angles, one at each vertex, bounded by a pair of adjacent sides. The same question can be asked for other geometrical shapes. for example, angles of a quadrilateral are in the ratio 1:2:3:4 and we need to find the smallest or largest angle. This can be solved in the same manner as we did in our solution.