If C is the midpoint of the point `A(-3,-4) and B(-5,6),` then AC =

Options

A. `5`
B. `sqrt(26)`
C. `sqrt(61)`
D. 8

Answer

To find the length of AC, where C is the midpoint of points A and B, we can follow these steps:

### Step 1: Identify the coordinates of points A and B
The coordinates of point A are given as A(-3, -4) and the coordinates of point B are given as B(-5, 6).

### Step 2: Use the midpoint formula to find the coordinates of point C
The midpoint C of two points A(x1, y1) and B(x2, y2) is calculated using the formula:
\[ C\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Substituting the coordinates of A and B:
\[ C\left(\frac{-3 + (-5)}{2}, \frac{-4 + 6}{2}\right) = C\left(\frac{-8}{2}, \frac{2}{2}\right) = C(-4, 1) \]

### Step 3: Calculate the length of segment AB using the distance formula
The distance between two points A(x1, y1) and B(x2, y2) is given by the formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of A and B:
\[ AB = \sqrt{(-5 - (-3))^2 + (6 - (-4))^2} = \sqrt{(-5 + 3)^2 + (6 + 4)^2} \]
This simplifies to:
\[ AB = \sqrt{(-2)^2 + (10)^2} = \sqrt{4 + 100} = \sqrt{104} \]

### Step 4: Simplify the distance AB
We can simplify \(\sqrt{104}\):
\[ \sqrt{104} = \sqrt{4 \times 26} = \sqrt{4} \times \sqrt{26} = 2\sqrt{26} \]

### Step 5: Find the length of AC
Since C is the midpoint of AB, the length of AC is half of AB:
\[ AC = \frac{AB}{2} = \frac{2\sqrt{26}}{2} = \sqrt{26} \]

### Final Answer
Thus, the length of AC is \(\sqrt{26}\).

---