Problem 57 - Algebra and Trigonometry

Question

Sketch the triangle with the given vertices, and use a determinant to find its area. $$ (-1,3),(2,9),(5,-6) $$

Step-by-Step Solution

Answer

The area of the triangle is 31.5 square units.

Step 1: Plot the Points

Plot the vertices of the triangle on a coordinate plane. The vertices given are \((-1, 3)\), \((2, 9)\), and \((5, -6)\). When you plot these points, you will see a triangle formed by connecting them.

Step 2: Use the Formula for Area

The area of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can be found using the formula: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \]. This formula is derived from the determinant of a matrix representing the triangle points.

Step 3: Substitute the Coordinates

Substitute \((x_1, y_1) = (-1, 3)\), \((x_2, y_2) = (2, 9)\), \((x_3, y_3) = (5, -6)\) into the formula: \[ \text{Area} = \frac{1}{2} \left| -1(9 + 6) + 2(-6 - 3) + 5(3 - 9) \right| \].

Step 4: Calculate the Determinant

Compute each term: - \(-1(9 + 6) = -1 \times 15 = -15\)- \(2(-6 - 3) = 2 \times (-9) = -18\)- \(5(3 - 9) = 5 \times (-6) = -30\).Add these values: \(-15 - 18 - 30 = -63\).

Step 5: Find the Absolute Value and Compute the Area

The absolute value of \(-63\) is \(63\). Therefore, the area is: \[ \text{Area} = \frac{1}{2} \times 63 = 31.5 \].

Key Concepts

Triangle Area

The area of a triangle can be calculated easily with a formula involving determinants. This formula is especially useful in coordinate geometry, where the vertices of the triangle are known. The area formula derived from determinants is: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \] where \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are the coordinates of the vertices of the triangle.

• The expression inside the absolute value signs calculates the determinant based on the coordinate differences.
• The division by 2 reflects the property of determinants that relate to the area of triangles.
• The absolute value ensures the area is always a positive quantity, aligning with its geometric interpretation.

This method is powerful as it generalizes to any set of vertices in the coordinate plane, allowing us to find the area without directly measuring height or base.

Coordinate Geometry

Coordinate geometry, also known as analytical geometry, involves using algebraic equations to describe and solve geometric problems. It brings together algebra and geometry, enabling us to explore shapes using a coordinate plane.

• In a coordinate plane, points are represented by coordinates \((x, y)\).
• Geometric shapes like triangles are defined by the positions of their vertices.
• The distances, angles, and areas can be calculated using these coordinate positions.

This branch is incredibly useful in computer graphics, physics, and engineering because it allows exact, formulaic representations of spatial ideas. Determinants serve as a great tool here as they simplify calculations related to areas and slopes in coordinate geometry.

Vertices

Vertices are pivotal points in geometry, representing the corners or intersection points of geometric shapes. In a triangle, these vertices are the three points where its sides meet.

• Each vertex of a triangle is represented by a pair of coordinates \((x, y)\) in the coordinate plane.
• The position of vertices determines the shape and size of the triangle.
• Interacting with vertices through transformations or translations affects the entire shape.

Understanding vertices in the context of coordinate geometry allows us to connect algebraic ways of expressing geometric properties, such as location, orientation, and area. By learning to manipulate these coordinates and apply formulas, you can efficiently solve complex problems across various fields.