Diagonals AC and BD of square ABCD intersect at P. Coordinates of points B and D are (9, -2) and (1, 6) respectively. (i) Find the coordinates of point P. (ii) Find the length of the side of the square.
Show Hint
Always remember that diagonals of a square are equal and perpendicular bisectors of each other.
Using the relation \( d = s\sqrt{2} \) directly gives the side length \( s \) without needing to use trigonometric functions or resolving the coordinates of vertices \( A \) and \( C \).
Step 1: Understanding the Question:
The topic of this question is Coordinate Geometry combined with the geometric properties of a Square.
We are given the coordinates of two opposite vertices of a square, \( B(9, -2) \) and \( D(1, 6) \).
The diagonals of a square bisect each other at right angles at their intersection point \( P \).
We need to find the coordinates of \( P \) (the midpoint of \( BD \)) and the length of the side of the square.
Step 2: Key Formula or Approach:
- Midpoint formula for coordinates:
\[ P(x, y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
- Distance formula to find the length of diagonal \( BD \):
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- Relationship between side \( s \) and diagonal \( d \) of a square:
\[ d = s\sqrt{2} \implies s = \frac{d}{\sqrt{2}} \]
Step 3: Detailed Explanation:
1. Part (i): Find coordinates of P:
Since \( P \) is the intersection point of the diagonals of square \( ABCD \), it is the midpoint of diagonal \( BD \).
Using the midpoint formula with \( B(9, -2) \) and \( D(1, 6) \):
\[ x_P = \frac{9 + 1}{2} = \frac{10}{2} = 5 \]
\[ y_P = \frac{-2 + 6}{2} = \frac{4}{2} = 2 \]
Thus, the coordinates of point \( P \) are \( (5, 2) \).
2. Part (ii): Find the length of the side of the square:
First, calculate the length of diagonal \( BD \) using the distance formula:
\[ BD = \sqrt{(1 - 9)^2 + (6 - (-2))^2} \]
\[ BD = \sqrt{(-8)^2 + (8)^2} \]
\[ BD = \sqrt{64 + 64} = \sqrt{128} \]
Simplify the radical:
\[ BD = \sqrt{64 \times 2} = 8\sqrt{2}\text{ units} \]
3. Use the relationship between the side and diagonal of a square:
Let the side length of the square be \( s \).
\[ s\sqrt{2} = 8\sqrt{2} \]
Divide both sides by \( \sqrt{2} \):
\[ s = 8\text{ units} \]
Step 4: Final Answer:
(i) The coordinates of point P are \((5, 2)\).
(ii) The length of the side of the square is 8 units.
