Geological Survey Triangle Area
The problem requires calculating the area of a triangular monitoring zone formed by three research stations with given coordinates: P(2,5), Q(8,1), and R(4,9).
The Shoelace Formula provides an efficient method for this calculation.
Shoelace Formula Coordinates Method
The formula for the area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is:
$ \text{Area} = \frac{1}{2} |(x_1y_2 + x_2y_3 + x_3y_1) - (y_1x_2 + y_2x_3 + y_3x_1)| $Identify the coordinates:
- P: $(x_1, y_1) = (2, 5)$
- Q: $(x_2, y_2) = (8, 1)$
- R: $(x_3, y_3) = (4, 9)$
Area Calculation Steps
- List coordinates sequentially and repeat the first: P(2,5), Q(8,1), R(4,9), P(2,5).
- Calculate the sum of the products along the downward diagonals:
$ (2 \times 1) + (8 \times 9) + (4 \times 5) = 2 + 72 + 20 = 94 $
- Calculate the sum of the products along the upward diagonals:
$ (5 \times 8) + (1 \times 4) + (9 \times 2) = 40 + 4 + 18 = 62 $
- Compute the area using the formula:
$ \text{Area} = \frac{1}{2} |(\text{Sum Downward}) - (\text{Sum Upward})| $
$ \text{Area} = \frac{1}{2} |94 - 62| = \frac{1}{2} |32| = 16 $
Monitoring Zone Area Result
The area of the triangular monitoring zone is 16 square kilometers.
Important Questions from Coordinate Geometry
Find the co-ordinates of the centroid of a triangle whose vertices are A(1, 4), B(7, 8) and C(10, 12).
- If x² + y² - 16x + 38y + 425 = 0, then the value of x² + y² is:
- If x² + y² - 12x + 18y + 117 = 0, then the value of x² + y² is:
- \(A(1, 2)\), \(B(a, -26)\), \(C(13, -14)\) and \(D(6, 14)\) are the vertices of a parallelogram, taken in order, find the value of \(a\).
- What would be the area of the triangle with \(A(8, 8)\), \(B(22, 8)\) and \(C(16, -10)\) as vertices?
