If the vertex of square OABC are `O(0,0),A(a, b),B(8,-6),C(c,d)` then `|a|+|b|=`

Answer

To solve the problem, we need to find the values of \(a\) and \(b\) such that the vertices of the square \(OABC\) are given as \(O(0,0)\), \(A(a,b)\), \(B(8,-6)\), and \(C(c,d)\). We will use the properties of squares and the coordinates of the given points.

### Step-by-Step Solution:

1. **Understanding the Properties of a Square**:
A square has all sides of equal length and the diagonals are equal and bisect each other at right angles.

2. **Finding the Length of Side \(OB\)**:
The length of side \(OB\) can be calculated using the distance formula:
\[ OB = \sqrt{(8 - 0)^2 + (-6 - 0)^2} = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]

3. **Finding the Coordinates of Point \(A\)**:
Since \(O\) and \(B\) are two vertices of the square, the coordinates of point \(A\) can be found using the fact that \(OA\) is perpendicular to \(OB\) and has the same length. The slope of line \(OB\) is:
\[ \text{slope of } OB = \frac{-6 - 0}{8 - 0} = -\frac{3}{4} \]
The slope of line \(OA\) will be the negative reciprocal of the slope of \(OB\):
\[ \text{slope of } OA = \frac{4}{3} \]

4. **Finding the Equation of Line \(OA\)**:
Using point-slope form, the equation of line \(OA\) through point \(O(0,0)\) is:
\[ y = \frac{4}{3}x \]

5. **Finding the Coordinates of Point \(A\)**:
Since \(OA\) is equal in length to \(OB\) (which is 10), we can set up the equation:
\[ \sqrt{(a - 0)^2 + (b - 0)^2} = 10 \]
This simplifies to:
\[ a^2 + b^2 = 100 \]

6. **Finding the Coordinates of Point \(C\)**:
The coordinates of point \(C\) can be found using the midpoint of \(OB\) and the fact that \(C\) is also a vertex of the square. The midpoint \(M\) of \(OB\) is:
\[ M = \left(\frac{0 + 8}{2}, \frac{0 - 6}{2}\right) = (4, -3) \]
The coordinates of point \(C\) can be determined by maintaining the perpendicular distance from \(M\) to \(C\) equal to the distance from \(M\) to \(A\).

7. **Finding \(|a| + |b|\)**:
Since we have \(a^2 + b^2 = 100\) and \(A\) lies on the line \(y = \frac{4}{3}x\), we can substitute \(b = \frac{4}{3}a\) into the equation:
\[ a^2 + \left(\frac{4}{3}a\right)^2 = 100 \]
This leads to:
\[ a^2 + \frac{16}{9}a^2 = 100 \implies \frac{25}{9}a^2 = 100 \implies a^2 = 36 \implies a = 6 \text{ or } -6 \]
Substituting \(a = 6\) into \(b = \frac{4}{3}a\):
\[ b = \frac{4}{3} \cdot 6 = 8 \]
Thus, \(|a| + |b| = |6| + |8| = 6 + 8 = 14\).

### Final Answer:
\[ |a| + |b| = 14 \]