Problem 50 - Algebra and Trigonometry

Question

Determinants are used to find the area of a triangle whose vertices are given by three points in a rectangular coordinate system. The area of a triangle with vertices \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\) and \(\left(x_{3}, y_{3}\right)\) is $$ \text { Area }=\pm \frac{1}{2}\left|\begin{array}{lll} {x_{1}} & {y_{1}} & {1} \\ {x_{2}} & {y_{2}} & {1} \\ {x_{3}} & {y_{3}} & {1} \end{array}\right| $$ where the \(\pm\) symbol indicates that the appropriate sign should be chosen to yield a positive area. Use this information to work Exercises \(49-50\). Use determinants to find the area of the triangle whose vertices are \((1,1),(-2,-3),\) and \((11,-3)\)

Step-by-Step Solution

Answer

The area of the triangle whose vertices are \((1,1),(-2,-3),\) and \((11,-3)\) is 26 units.

Step 1: Formula for Area

Write down the formula for finding area of triangle, which is \(Area= ±0.5 * [x1(y2-y3) + x2(y3-y1) + x3(y1-y2)]\). The absolute value is taken to ensure area is positive.

Step 2: Plug values into formula

Substitute given coordinates into the formula: \(Area= ±0.5 * [1(-3-(-3)) + (-2)(-3-1) + 11(1-(-3))]\).

Step 3: Simplify the equation

After plugging the values into the formula, simplify the equation. Here, the equation becomes \(Area = 0*(-6) -2*(-4) + 11*(4) = 0 + 8 + 44 = 52\).

Step 4: Final Calculation

The final calculation gives the value \(Area = 0.5 * 52 = 26 \).

Key Concepts

Determinants in Algebra

Determinants are a fundamental concept in algebra and are particularly useful in systems of linear equations, matrix theory, and vector spaces. In essence, a determinant is a scalar value that is computed from the elements of a square matrix. It provides important properties about the matrix, such as whether it is invertible and the volume scaling factor when the matrix is viewed as a linear transformation.

In the context of solving for the area of a triangle using coordinates, the determinant serves as a tool that simplifies the computation. We use a 3x3 matrix, with each row representing the x-coordinate, y-coordinate, and a fixed value of 1 for each point that defines the vertices of the triangle. The absolute value of the determinant of this matrix, multiplied by a half, gives us the area of the triangle. This method avoids the need for more complex geometric or trigonometric calculations, making determinants a powerful technique in coordinate geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometry using a coordinate system. This branch of mathematics allows us to describe geometric figures algebraically and solve geometric problems by employing algebraic equations. In the case of finding the area of a triangle, we translate the problem of spatial measurement into an algebraic form using the coordinates of its vertices.

The method shown in the exercise leverages the concept of determinants to calculate the area without having to measure any sides or angles physically. By applying the coordinates directly, we not only circumvent the need for trigonometrical solutions but also pave the way for solving more complex problems, such as finding the area of polygons or determining the intersection points of lines. The elegance of coordinate geometry lies in its ability to unify algebra and geometry, opening up a vast range of possibilities for mathematical analysis and application.

Trigonometry Area Calculation

In contrast to using determinants and coordinate geometry, trigonometry area calculation is another method used to find the area of a triangle. Trigonometry deals with the relationships between the angles and sides of triangles and is commonly associated with right-angled triangles. However, trigonometric functions and principles can also be applied to calculate the area of non-right-angled triangles.

The typical formula used in trigonometry to determine the area of a triangle is \(\frac{1}{2} \times base \times height\) or, in the case of oblique triangles, \(\frac{1}{2} \times ab \times \text{sin}(C)\), where 'a' and 'b' are the lengths of two sides of the triangle, and 'C' is the included angle. While trigonometry provides a precise solution, it often necessitates additional information such as the lengths of sides and measurements of angles, which may not always be readily available. This makes the determinant method showcased in the exercise particularly useful for scenarios where the vertex coordinates are known, but other details of the triangle are not.