What is the difference between distnace of mid point of points `( - 3,5) and (7, -6)` from x-axis y-axis ?

Options

A. `5/2`
B. `3/2`
C. `5/4`
D. `3/4`

Answer

To solve the problem step by step, we need to find the midpoint of the points (-3, 5) and (7, -6), then calculate the distances of this midpoint from the x-axis and the y-axis, and finally find the difference between these two distances.

### Step 1: Find the Midpoint
The formula for the midpoint \( M(x, y) \) of two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Using the points (-3, 5) and (7, -6):
- \( x_1 = -3 \), \( y_1 = 5 \)
- \( x_2 = 7 \), \( y_2 = -6 \)

Calculating the midpoint:
\[ M_x = \frac{-3 + 7}{2} = \frac{4}{2} = 2 \]
\[ M_y = \frac{5 + (-6)}{2} = \frac{-1}{2} \]

So, the midpoint \( M \) is \( (2, -\frac{1}{2}) \).

### Step 2: Calculate the Distance from the x-axis
The distance from a point \( (x, y) \) to the x-axis is given by the absolute value of the y-coordinate:
\[ \text{Distance from x-axis} = |M_y| = \left| -\frac{1}{2} \right| = \frac{1}{2} \]

### Step 3: Calculate the Distance from the y-axis
The distance from a point \( (x, y) \) to the y-axis is given by the absolute value of the x-coordinate:
\[ \text{Distance from y-axis} = |M_x| = |2| = 2 \]

### Step 4: Find the Difference Between the Distances
Now, we need to find the difference between the distance from the x-axis and the distance from the y-axis:
\[ \text{Difference} = \text{Distance from y-axis} - \text{Distance from x-axis} = 2 - \frac{1}{2} \]

To perform the subtraction:
\[ 2 = \frac{4}{2} \quad \text{(converting 2 to a fraction with a common denominator)} \]
\[ \text{Difference} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2} \]

### Final Answer
The difference between the distance of the midpoint from the x-axis and the y-axis is \( \frac{3}{2} \).

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