If A(5,8),B(-3,4) and C(7,k) are vertices of `triangleABC` and `m angleB = 90^(@)`, then k=

Options

A. 16
B. `-12`
C. `-16`
D. 12

Answer

To find the value of \( k \) such that angle \( B \) in triangle \( ABC \) is \( 90^\circ \), we will use the Pythagorean theorem. The coordinates of the points are given as follows: \( A(5, 8) \), \( B(-3, 4) \), and \( C(7, k) \).

### Step 1: Use the Pythagorean theorem
Since \( \angle B = 90^\circ \), we can apply the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]

### Step 2: Calculate \( AB^2 \)
Using the distance formula, the distance \( AB \) is calculated as:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points \( A \) and \( B \):
\[ AB = \sqrt{((-3) - 5)^2 + (4 - 8)^2} = \sqrt{(-8)^2 + (-4)^2} = \sqrt{64 + 16} = \sqrt{80} \]
Thus,
\[ AB^2 = 80 \]

### Step 3: Calculate \( BC^2 \)
Now, we calculate the distance \( BC \):
\[ BC = \sqrt{(7 - (-3))^2 + (k - 4)^2} = \sqrt{(7 + 3)^2 + (k - 4)^2} = \sqrt{10^2 + (k - 4)^2} = \sqrt{100 + (k - 4)^2} \]
Thus,
\[ BC^2 = 100 + (k - 4)^2 \]

### Step 4: Calculate \( AC^2 \)
Next, we calculate the distance \( AC \):
\[ AC = \sqrt{(7 - 5)^2 + (k - 8)^2} = \sqrt{(2)^2 + (k - 8)^2} = \sqrt{4 + (k - 8)^2} \]
Thus,
\[ AC^2 = 4 + (k - 8)^2 \]

### Step 5: Set up the equation
Now we can set up the equation using the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
Substituting the values we found:
\[ 4 + (k - 8)^2 = 80 + (100 + (k - 4)^2) \]

### Step 6: Simplify the equation
Expanding both sides:
\[ 4 + (k^2 - 16k + 64) = 80 + 100 + (k^2 - 8k + 16) \]
This simplifies to:
\[ k^2 - 16k + 68 = 180 + k^2 - 8k + 16 \]
Now, cancel \( k^2 \) from both sides:
\[ -16k + 68 = 196 - 8k \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ -16k + 8k = 196 - 68 \]
\[ -8k = 128 \]
\[ k = -16 \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{-16} \]