in the triangle `ABC , /_A=x^@,/_B=3x^@,/_C=y^@`. if `3y-5x=30` then prove that the triangle is right angled.

Answer

To prove that triangle ABC is a right-angled triangle given the angles and the equation \(3y - 5x = 30\), we will follow these steps:

### Step 1: Express the angles in terms of \(x\) and \(y\)
Given:
- Angle \(A = x^\circ\)
- Angle \(B = 3x^\circ\)
- Angle \(C = y^\circ\)

### Step 2: Use the property of the sum of angles in a triangle
The sum of the angles in triangle ABC is:
\[ A + B + C = 180^\circ \]
Substituting the values of the angles:
\[ x + 3x + y = 180^\circ \]
This simplifies to:
\[ 4x + y = 180^\circ \quad \text{(Equation 1)} \]

### Step 3: Solve the given equation for \(y\)
We are given:
\[ 3y - 5x = 30 \]
Rearranging this gives:
\[ 3y = 30 + 5x \]
Dividing by 3:
\[ y = \frac{30 + 5x}{3} \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 2 into Equation 1
Now, substitute \(y\) from Equation 2 into Equation 1:
\[ 4x + \frac{30 + 5x}{3} = 180 \]
To eliminate the fraction, multiply the entire equation by 3:
\[ 3(4x) + (30 + 5x) = 540 \]
This simplifies to:
\[ 12x + 30 + 5x = 540 \]
Combining like terms:
\[ 17x + 30 = 540 \]

### Step 5: Solve for \(x\)
Subtract 30 from both sides:
\[ 17x = 510 \]
Now, divide by 17:
\[ x = 30 \]

### Step 6: Find \(y\)
Substituting \(x = 30\) back into Equation 2 to find \(y\):
\[ y = \frac{30 + 5(30)}{3} = \frac{30 + 150}{3} = \frac{180}{3} = 60 \]

### Step 7: Determine the angles
Now we can find the angles:
- Angle \(A = x = 30^\circ\)
- Angle \(B = 3x = 3(30) = 90^\circ\)
- Angle \(C = y = 60^\circ\)

### Step 8: Conclusion
Since angle \(B = 90^\circ\), triangle ABC is a right-angled triangle.