Hey! To find the area of APOQ, we need to determine the coordinates of points P and Q first. Since P and Q are 5 units away from the origin, we can use this information to find their coordinates.
The equation of the line x - y = 1 can be rearranged as y = x - 1.
To find point P, we substitute x = 5 into the equation:
y = 5 - 1 = 4
Therefore, P is located at (5, 4).
To find point Q, we substitute x = -5 into the equation:
y = -5 - 1 = -6
So, Q is located at (-5, -6).
Now that we have the coordinates of P (5, 4), O (0, 0), and Q (-5, -6), we can calculate the area of APOQ. The area of a triangle can be found using the formula:
Area = 1/2 * base * height.
The base of APOQ is the distance between points P and Q, which is 5 units. The height is the perpendicular distance from point O to the line segment PQ. Since the line x - y = 1 is perpendicular to the line connecting points P and Q, the height can be found by calculating the perpendicular distance from the origin to the line.
Using the formula for the distance between a point and a line, the perpendicular distance from the origin to the line x - y = 1 is:
Distance = |0 - 1| / √(1^2 + (-1)^2) = 1 / √2 = √2 / 2.
Therefore, the height of APOQ is √2 / 2.
Now we can calculate the area:
Area = 1/2 * base * height = 1/2 * 5 * (√2 / 2) = 5√2 / 4.
So, the area of APOQ is 5√2 / 4.