The difference between the largest and the smallest angles of a triangle whose angles are in the ratio of `5:3:10` is

Options

A. `20^@`
B. `30^@`
C. `50^@`
D. `70^@`

Answer

To find the difference between the largest and smallest angles of a triangle whose angles are in the ratio of 5:3:10, follow these steps:

### Step 1: Define the angles in terms of a variable
Let the angles of the triangle be represented as:
- Angle A = 5x
- Angle B = 3x
- Angle C = 10x

### Step 2: Identify the largest and smallest angles
From the ratios, we can see that:
- Angle C (10x) is the largest angle.
- Angle B (3x) is the smallest angle.

### Step 3: Calculate the difference between the largest and smallest angles
The difference between the largest angle (Angle C) and the smallest angle (Angle B) is:
\[ \text{Difference} = \text{Angle C} - \text{Angle B} = 10x - 3x = 7x \]

### Step 4: Find the sum of the angles
The sum of the angles in a triangle is always 180 degrees. Therefore:
\[ \text{Sum} = \text{Angle A} + \text{Angle B} + \text{Angle C} = 5x + 3x + 10x = 18x \]
Setting this equal to 180 degrees gives:
\[ 18x = 180 \]

### Step 5: Solve for x
To find x, divide both sides by 18:
\[ x = \frac{180}{18} = 10 \]

### Step 6: Substitute x back to find the angles
Now, substitute x back to find the actual angles:
- Angle A = 5x = 5(10) = 50 degrees
- Angle B = 3x = 3(10) = 30 degrees
- Angle C = 10x = 10(10) = 100 degrees

### Step 7: Calculate the difference in degrees
Now, we can calculate the difference in degrees:
\[ \text{Difference} = \text{Angle C} - \text{Angle B} = 100 - 30 = 70 \text{ degrees} \]

### Final Answer
The difference between the largest and smallest angles of the triangle is **70 degrees**.

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