A triangle ABC right angled at A has points A and B as (2, 3) and (0, -1) respectively. If BC = 5 units, then the point C is

Options

A. (4, 2)
B. (-4, 2)
C. (-4, 4)
D. (4, -4)

Answer

To find the coordinates of point C in triangle ABC, where A is (2, 3), B is (0, -1), and BC = 5 units, we can follow these steps:

### Step 1: Understand the triangle configuration
Since triangle ABC is right-angled at A, we can visualize the triangle with A at (2, 3) and B at (0, -1). Point C will be somewhere such that the distance BC is 5 units.

### Step 2: Use the distance formula for BC
The distance between points B (0, -1) and C (a, b) can be calculated using the distance formula:
\[ BC = \sqrt{(a - 0)^2 + (b - (-1))^2} = 5 \]
This simplifies to:
\[ \sqrt{a^2 + (b + 1)^2} = 5 \]

### Step 3: Square both sides to eliminate the square root
Squaring both sides gives:
\[ a^2 + (b + 1)^2 = 25 \]

### Step 4: Expand the equation
Expanding the equation:
\[ a^2 + (b^2 + 2b + 1) = 25 \]
This simplifies to:
\[ a^2 + b^2 + 2b + 1 = 25 \]
\[ a^2 + b^2 + 2b - 24 = 0 \quad \text{(Equation 1)} \]

### Step 5: Find the slope of AB
Next, we need to find the slopes of sides AB and AC. The slope of line AB can be calculated as:
\[ \text{slope of AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 3}{0 - 2} = \frac{-4}{-2} = 2 \]

### Step 6: Find the slope of AC
Let the slope of AC be \(m_{AC}\). Since triangle ABC is right-angled at A, the product of the slopes of AB and AC must equal -1:
\[ 2 \cdot m_{AC} = -1 \]
Thus,
\[ m_{AC} = -\frac{1}{2} \]

### Step 7: Write the equation for slope AC
Using the slope formula for AC:
\[ \text{slope of AC} = \frac{b - 3}{a - 2} = -\frac{1}{2} \]
Cross-multiplying gives:
\[ 2(b - 3) = -1(a - 2) \]
This simplifies to:
\[ 2b - 6 = -a + 2 \]
Rearranging gives:
\[ a + 2b - 8 = 0 \quad \text{(Equation 2)} \]

### Step 8: Solve the system of equations
Now we have two equations:
1. \(a^2 + b^2 + 2b - 24 = 0\)
2. \(a + 2b - 8 = 0\)

From Equation 2, we can express \(a\) in terms of \(b\):
\[ a = 8 - 2b \]

### Step 9: Substitute into Equation 1
Substituting \(a\) into Equation 1:
\[ (8 - 2b)^2 + b^2 + 2b - 24 = 0 \]
Expanding this:
\[ 64 - 32b + 4b^2 + b^2 + 2b - 24 = 0 \]
Combining like terms gives:
\[ 5b^2 - 30b + 40 = 0 \]
Dividing through by 5:
\[ b^2 - 6b + 8 = 0 \]

### Step 10: Factor the quadratic
Factoring gives:
\[ (b - 2)(b - 4) = 0 \]
Thus, \(b = 2\) or \(b = 4\).

### Step 11: Find corresponding values of a
1. If \(b = 2\):
\[ a = 8 - 2(2) = 4 \]
So, one possible point C is (4, 2).

2. If \(b = 4\):
\[ a = 8 - 2(4) = 0 \]
So, another possible point C is (0, 4).

### Conclusion
The possible coordinates for point C are (4, 2) and (0, 4). Since the options provided include (4, 2), we can conclude that the point C is:
\[ \text{C} = (4, 2) \]