In right angled `Delta ABC, angle B=90^(@) and AB=sqrt34` units . The co-ordinares of points B, C are (4. 2) and (-1, y) respectively . If ar ` Delta ABC=17` sq . Units, then find the value of y .

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the given information
We have a right-angled triangle ABC with:
- Angle B = 90 degrees
- Length of AB = √34 units
- Coordinates of B = (4, 2)
- Coordinates of C = (-1, y)
- Area of triangle ABC = 17 square units

### Step 2: Use the area formula for a triangle
The area of a triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, we can take BC as the base and AB as the height. Therefore, we can write:
\[ \frac{1}{2} \times BC \times AB = 17 \]
Substituting the value of AB:
\[ \frac{1}{2} \times BC \times \sqrt{34} = 17 \]

### Step 3: Solve for BC
Multiplying both sides by 2:
\[ BC \times \sqrt{34} = 34 \]
Now, divide both sides by √34:
\[ BC = \frac{34}{\sqrt{34}} = \sqrt{34} \]

### Step 4: Find the length of BC using the distance formula
The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Applying this to points B (4, 2) and C (-1, y):
\[ BC = \sqrt{((-1) - 4)^2 + (y - 2)^2} \]
This simplifies to:
\[ BC = \sqrt{(-5)^2 + (y - 2)^2} = \sqrt{25 + (y - 2)^2} \]

### Step 5: Set the two expressions for BC equal
Now we have two expressions for BC:
\[ \sqrt{25 + (y - 2)^2} = \sqrt{34} \]
Squaring both sides:
\[ 25 + (y - 2)^2 = 34 \]

### Step 6: Solve for y
Subtract 25 from both sides:
\[ (y - 2)^2 = 9 \]
Taking the square root of both sides gives:
\[ y - 2 = 3 \quad \text{or} \quad y - 2 = -3 \]
Thus:
1. \(y - 2 = 3 \Rightarrow y = 5\)
2. \(y - 2 = -3 \Rightarrow y = -1\)

### Step 7: Conclusion
The possible values for y are:
\[ y = 5 \quad \text{or} \quad y = -1 \]