Solving Geometry Problem: Coordinates and Properties of Triangle PQR

The question at hand is to determine the coordinates of point A in the context of triangle PQR, given that P, Q, and R are three points forming the sides of the triangle, and B is the foot of the perpendicular from Q to PR. Let us delve into the details to solve this problem step by step.

Given Information and Calculations

Side lengths

PQ sqrt(3^2 4^2) 5

QR sqrt(1^2 5^2) sqrt(26) ≈ 5.0990

PR sqrt(2^2 1^2) sqrt(5) ≈ 2.2361

The semi-perimeter s of the triangle is calculated as:

s (5 5.0990 2.2361) / 2 ≈ 6.1675

The area of the triangle is found using Heron's formula:

Area sqrt(s(s - a)(s - b)(s - c)) ≈ sqrt(6.1675 * (6.1675 - 5) * (6.1675 - 5.0990) * (6.1675 - 2.2361)) ≈ 5.4998

Altitude Calculation

The altitude QB from Q to PR is calculated as:

QB 2 * Area / PR ≈ 2 * 5.4998 / 2.2361 ≈ 4.9191

Addressing the Question: Coordinates of Point A

The crux of the problem lies in explaining the origin of point A. In the original problem statement, there is no mention of A. Therefore, to resolve the ambiguity, we need to consider the specific context of the problem. Given the information above, we can infer that point A is not a point within the triangle but possibly a point related to the geometric properties or configuration.

One possible scenario could be that point A is an additional point in the plane that interacts with the given triangle. For this, we need further details or a more complete problem statement. Without specific coordinates for point A, it is challenging to determine its exact location.

However, if we assume a common type of problem where A could be a point on the extended line of one of the sides or a point of intersection, we can explore such possibilities further.

For example, if A is a point on the extension of PR or a point related to the perpendicularity and other geometric properties, the coordinates could be derived based on the given vector information and additional geometric rules.

Conclusion

While the original problem statement does not explicitly define the coordinates of point A, the solution process provides a framework for detailed geometric analysis. Further clarification or additional problem details would be needed to precisely determine the coordinates of A.

To summarize, the key steps involved are:

Calculating side lengths and the semi-perimeter. Determining the area using Heron's formula. Calculating the altitude based on the area and the side length.

Note: The coordinates of A could vary based on additional geometric relationships and conditions not provided in the initial problem statement.