The angles of a triangle are in the ratio 5 : 3 : 7. Then triangle is :

Question

Accepted Answer

The given problem states that the angles of a triangle are in the ratio 5:3:7. To determine the nature of the triangle, we first need to find the actual measures of the angles.

1. **Calculating the Angles:**
   - Let the angles be $5x$, $3x$, and $7x$.
   - The sum of the angles in a triangle is always $180^\circ$.
   - Therefore, we have the equation: 
     $$
     5x + 3x + 7x = 180
     $$
   - Simplifying, we get:
     $$
     15x = 180
     $$
   - Solving for $x$, we find:
     $$
     x = \frac{180}{15} = 12
     $$

2. **Finding the Angles:**
   - The angles are $5x = 60^\circ$, $3x = 36^\circ$, and $7x = 84^\circ$.

3. **Justifying Option 1: An Acute Angled Triangle:**
   - An acute angled triangle is one where all the angles are less than $90^\circ$.
   - In this case, all angles $60^\circ$, $36^\circ$, and $84^\circ$ are indeed less than $90^\circ$.
   - Therefore, the triangle is an acute angled triangle, supporting Option 1 as the correct answer.

4. **Explaining Why Other Options Are Incorrect:**
   - **Option 2: An Obtuse Angled Triangle:**
     - An obtuse angled triangle has one angle greater than $90^\circ$.
     - Since all angles in this triangle are less than $90^\circ$, it cannot be obtuse.
   
   - **Option 3: A Right Angled Triangle:**
     - A right angled triangle has one angle exactly $90^\circ$.
     - None of the angles in this triangle is $90^\circ$, so it cannot be a right angled triangle.
   
   - **Option 4: An Isosceles Triangle:**
     - An isosceles triangle has at least two equal angles.
     - The angles $60^\circ$, $36^\circ$, and $84^\circ$ are all different, so the triangle is not isosceles.

In conclusion, based on the calculations and definitions, the triangle with angles in the ratio 5:3:7 is an acute angled triangle, confirming that Option 1 is indeed the correct answer.