1. **Problem statement:** We are given two points A and B, with B at coordinates (4, 6). The line L is the perpendicular bisector of the segment AB, and its equation is given as $$y = -2x + 9$$. We need to find the coordinates of point A. 2. **Key concepts:** - The perpendicular bisector of a segment passes through the midpoint of the segment. - It is perpendicular to the segment AB. 3. **Step 1: Find the midpoint M of AB.** Let A have coordinates $ (x, y) $. Then the midpoint M is: $$ M = \left( \frac{x + 4}{2}, \frac{y + 6}{2} \right) $$ 4. **Step 2: Since L is the perpendicular bisector, M lies on L.** Substitute the midpoint coordinates into the line equation: $$ \frac{y + 6}{2} = -2 \cdot \frac{x + 4}{2} + 9 $$ Multiply both sides by 2: $$ y + 6 = -2(x + 4) + 18 $$ Simplify the right side: $$ y + 6 = -2x - 8 + 18 $$ $$ y + 6 = -2x + 10 $$ Subtract 6 from both sides: $$ y = -2x + 4 $$ 5. **Step 3: Use the slope condition for perpendicularity.** The slope of line L is $-2$. The segment AB is perpendicular to L, so its slope is the negative reciprocal: $$ m_{AB} = \frac{1}{2} $$ 6. **Step 4: Write the slope formula for AB.** $$ m_{AB} = \frac{6 - y}{4 - x} = \frac{1}{2} $$ Cross-multiply: $$ 2(6 - y) = 4 - x $$ $$ 12 - 2y = 4 - x $$ Rearranged: $$ -2y + x = 4 - 12 $$ $$ -2y + x = -8 $$ 7. **Step 5: Substitute $y$ from step 4 into this equation.** From step 4, we have: $$ y = -2x + 4 $$ Substitute into the equation: $$ -2(-2x + 4) + x = -8 $$ Simplify: $$ 4x - 8 + x = -8 $$ $$ 5x - 8 = -8 $$ Add 8 to both sides: $$ 5x = 0 $$ $$ x = 0 $$ 8. **Step 6: Find y using $x=0$.** $$ y = -2(0) + 4 = 4 $$ **Final answer:** The coordinates of point A are $$\boxed{(0, 4)}$$.