Sure! Let's solve this step-by-step. We're given that Harrison's house is halfway between the library and the town hall, and we have their coordinates:
- Library (L): [tex]\((6, -11)\)[/tex]
- Harrison's house (H): [tex]\((9, -3)\)[/tex]
- Town Hall (T): [tex]\((x, y)\)[/tex]
Given that Harrison's house is the midpoint, we can use the midpoint formula. The formula for the midpoint [tex]\(M\)[/tex] of two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\][/tex]
Here, [tex]\(H\)[/tex] is the midpoint, so:
[tex]\[H = \left(\frac{L_x + T_x}{2}, \frac{L_y + T_y}{2}\right)\][/tex]
Given [tex]\(L = (6, -11)\)[/tex] and [tex]\(H = (9, -3)\)[/tex], we need to solve for [tex]\(T = (x, y)\)[/tex].
Let's break it down:
1. The x-coordinate of the midpoint formula:
[tex]\[9 = \frac{6 + x}{2}\][/tex]
To isolate [tex]\(x\)[/tex], multiply both sides by 2:
[tex]\[18 = 6 + x\][/tex]
Subtract 6 from both sides:
[tex]\[x = 12\][/tex]
2. The y-coordinate of the midpoint formula:
[tex]\[-3 = \frac{-11 + y}{2}\][/tex]
To isolate [tex]\(y\)[/tex], multiply both sides by 2:
[tex]\[-6 = -11 + y\][/tex]
Add 11 to both sides:
[tex]\[y = 5\][/tex]
So, the coordinates of the town hall are:
[tex]\[T = (12, 5)\][/tex]
Therefore, the location of the town hall is [tex]\((12, 5)\)[/tex].
