The triangle with vertices A (2, 7), B (4, y) and `C(-2, 6)` is right angled at A if

Options

A. `y=-1`
B. y = 0
C. y = 1
D. none of these

Answer

To determine the value of \( y \) such that the triangle with vertices \( A(2, 7) \), \( B(4, y) \), and \( C(-2, 6) \) is right-angled at \( A \), we can follow these steps:

### Step 1: Understand the condition for right-angled triangles
A triangle is right-angled at a vertex if the slopes of the two lines forming the sides at that vertex are negative reciprocals of each other. Specifically, if the slopes of lines \( AB \) and \( AC \) are \( m_1 \) and \( m_2 \), then:
\[ m_1 \cdot m_2 = -1 \]

### Step 2: Calculate the slope of line \( AB \)
The slope \( m_{AB} \) of line \( AB \) can be calculated using the formula:
\[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{y - 7}{4 - 2} = \frac{y - 7}{2} \]

### Step 3: Calculate the slope of line \( AC \)
The slope \( m_{AC} \) of line \( AC \) is calculated as:
\[ m_{AC} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 7}{-2 - 2} = \frac{-1}{-4} = \frac{1}{4} \]

### Step 4: Set up the equation using the slopes
Since the triangle is right-angled at \( A \), we have:
\[ m_{AB} \cdot m_{AC} = -1 \]
Substituting the slopes we calculated:
\[ \left(\frac{y - 7}{2}\right) \cdot \left(\frac{1}{4}\right) = -1 \]

### Step 5: Solve for \( y \)
Now, we can solve the equation:
\[ \frac{y - 7}{2} \cdot \frac{1}{4} = -1 \]
Multiplying both sides by \( 8 \) (to eliminate the fractions):
\[ y - 7 = -8 \]
Adding \( 7 \) to both sides gives:
\[ y = -8 + 7 = -1 \]

### Final Answer
The value of \( y \) is \( -1 \).

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