Question

The measures of the three angles of a triangle are in the ratio 1 : 2 : 3. Then the triangle is:-
a)Right angled triangle
b)Equilateral triangle
c)Isosceles triangle
d)Obtuse angled triangle

Answer 1

Hint:- Before solving this question, we must know about different types of triangles. Here are some of them:-
Right – Angled Triangle: This is a triangle that is classified on the basis of the measurement of its angles. If the measure of any one of the angles of a triangle is 90°, then that triangle is said to be a right – angled triangle.
Equilateral Triangle: This is a triangle that is classified on the basis of the length of its sides. If the length of all the sides of a triangle is equal, then it is an equilateral triangle.
Isosceles Triangle: This is a triangle that is classified on the basis of the length of its sides. If the length of any two of the three sides of a triangle is equal, then it is an isosceles triangle.
Obtuse – Angled Triangle: This is a triangle that is classified on the basis of the measurement of its angles. If the measure of any one of the angles of a triangle is more than 90°, then that triangle is said to be an obtuse – angled triangle.

Complete step-by-step answer:
Let us now solve this question.
For the solution of this question, we must know about the angle sum property of triangles.
It states that the sum of the measures of all the three angles of a triangle must be 180°.
In other words, we can say that when we add all the measures of the three angles of any triangle, then the sum we should get must be equal to 180° always.
So, according to the question, as the measures of the three angles of a triangle are in the ratio of 1 : 2 : 3, we can form an equation by this.
We will take the measure of the three angles of this triangle to be \[1x,\ \ 2x\ \ and\ \ 3x\] .
We can equate \[1x,\ \ 2x\ \ and\ \ 3x\] to 180°.
So, the equation that we get is:-
\[1x\ \ +\ \ 2x\ \ +\ \ 3x\ \ =\ \ 180{}^\circ \]
We shall now solve this equation.
\[\begin{align}
  & 1x\ \ +\ \ 2x\ \ +\ \ 3x\ \ =\ \ 180{}^\circ \\
 & 6x\ \ =\ \ 180{}^\circ \\
 & x\ \ =\ \ \dfrac{180}{6} \\
 & x\ \ =\ \ 30{}^\circ \\
\end{align}\]
So, as we have calculated the value of ‘x’, we can find the measures of all the three angles now.
\[\begin{align}
  & x\ \ =\ \ 30{}^\circ \\
 & 1x\ \ =\ \ x\ \times \ 1\ \ =\ \ 30{}^\circ \ \times \ 1\ \ =\ \ 30{}^\circ \\
 & 2x\ \ =\ \ x\ \times \ 2\ \ =\ \ 30{}^\circ \ \times \ 2\ \ =\ \ 60{}^\circ \\
 & 3x\ \ =\ \ x\ \times \ 3\ \ =\ \ 30{}^\circ \ \times \ 3\ \ =\ \ 90{}^\circ \\
\end{align}\]
So, the measurement of the three angles is 30°, 60° and 90°.
Now, we can see that there is an angle of 90° in these three angles of a triangle. This shows that the measure of one of the angles of this triangle is 90°.
As mentioned in the hint provided above, it is written that if the measure of any one of the angles of a triangle is 90°, then that triangle is said to be a right – angled triangle.
Therefore, this triangle is a right – angled triangle.
Hence, option (a) Right – angled triangle is the correct one.

Note:-Let us now learn about acute angled triangle and scalene triangle.
Acute – Angled Triangle: This is a triangle that is classified on the basis of the measurement of its angles. If the measure of all the angles of a triangle is less than 90°, then that triangle is said to be an acute– angled triangle.
Scalene Triangle: This is a type triangle that is classified on the basis of the length of its sides. If the length of all the sides of a triangle is unequal, i.e. the length of each side is different, then it is a scalene triangle.