Problem 60 - Algebra and Trigonometry

Question

(a) If three points lie on a line, what is the area of the "triangle" that they determine? Use the answer to this question, together with the determinant formula for the area of a triangle, to explain why the points \(\left(a_{1}, b_{1}\right)\) \(\left(a_{2}, b_{2}\right),\) and \(\left(a_{3}, b_{3}\right)\) are collinear if and only if $$ \left|\begin{array}{lll}{a_{1}} & {b_{1}} & {1} \\ {a_{2}} & {b_{2}} & {1} \\\ {a_{3}} & {b_{3}} & {1}\end{array}\right|=0 $$ (b) Use a determinant to check whether each set of points is collinear. Graph them to verify your answer. (i) \((-6,4),(2,10),(6,13)\) (ii) \((-5,10),(2,6),(15,-2)\)

Step-by-Step Solution

Answer

(i) Points are collinear; (ii) Points are not collinear.

Step 1: Understand Collinear Points

If three points lie on a line, they do not form a triangle. This means the area of the triangle formed by these points is zero.

Step 2: Area of a Triangle Using Determinants

The area of a triangle with vertices at \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) is given by the formula: \[ \text{Area} = \frac{1}{2} \left| \begin{array}{ccc} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{array} \right| \]When points are collinear, the area is zero, leading to a determinant of zero.

Step 3: Apply the Determinant Formula for Collinearity

To determine if the points \( \left(a_{1}, b_{1}\right), \left(a_{2}, b_{2}\right), \left(a_{3}, b_{3}\right) \) are collinear, calculate the determinant: \[ \left| \begin{array}{ccc} a_1 & b_1 & 1 \ a_2 & b_2 & 1 \ a_3 & b_3 & 1 \end{array} \right| = 0 \]If this determinant is zero, the points are collinear.

Step 4: Evaluate Det1 for Points (-6,4), (2,10), (6,13)

Calculate the determinant: \[ \left| \begin{array}{ccc} -6 & 4 & 1 \ 2 & 10 & 1 \ 6 & 13 & 1 \end{array} \right| = (-6)(10)(1) + (4)(1)(6) + (1)(2)(13) - (1)(10)(6) - (4)(2)(1) - (-6)(13)(1) \]\[ = -60 + 24 + 26 - 60 - 8 + 78 \]\[ = 0 \]Thus, the points are collinear.

Step 5: Evaluate Det2 for Points (-5,10), (2,6), (15,-2)

Calculate the determinant: \[ \left| \begin{array}{ccc} -5 & 10 & 1 \ 2 & 6 & 1 \ 15 & -2 & 1 \end{array} \right| = (-5)(6)(1) + (10)(1)(15) + (1)(2)(-2) - (1)(6)(15) - (10)(2)(1) - (-5)(-2)(1) \]\[ = -30 + 150 - 4 - 90 - 20 - 10 \]\[ = -4 \]Thus, the points are not collinear.

Step 6: Verify with Graphs

Plot each set of points on a graph. For the first set, the points lie on a straight line, confirming they are collinear. For the second set, the points do not lie on a straight line, confirming they are not collinear.

Key Concepts

Determinants

Determinants are mathematical tools used to obtain various properties of matrices and systems of linear equations. In the context of geometric concepts, a determinant can help determine whether three given points in a plane are collinear. Specifically, it involves using a \(3 \times 3\) matrix that incorporates the coordinates of the three points.
When you write the points \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) into a determinant matrix like this:

• \(|\begin{matrix} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{matrix}|\).

The determinant of this matrix being zero signifies that the points are collinear. This is because the area of the triangle formed by these points would be zero if they lie on the same line.
Collinearity using determinants is a practical application, making it easier to decide if points share a single linear alignment without needing to visually check.

Area of a Triangle

Triangles are one of the simplest geometric shapes that can be described in the coordinate plane using their vertices. Each pair of vertices defines one of the triangle’s edges, and the entire set defines its area. Interestingly, you can calculate the area using determinants.
The formula to find the area of a triangle using its vertex coordinates \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) is:

• \( \text{Area} = \frac{1}{2} \left|\begin{matrix} x_1 & y_1 & 1 \ x_2 & y_2 & 1 \ x_3 & y_3 & 1 \end{matrix}\right| \).

This determinant-based formula stems from the use of a \(3 \times 3\) matrix, and the absolute value of that determinant gives twice the area of the triangle if the points are non-collinear.
Thus, if the determinant equals zero, the area is zero, indicating that the vertices do not enclose any space, proving collinearity.

Geometric Concepts

Understanding geometric concepts requires familiarity with basic shapes and their properties. Essential geometric figures like triangles, lines, and points are central to determining collinearity and shape areas.

Collinearity in geometric terms refers to multiple points lying on a single straight line. This means if points A, B, and C are collinear, point B lies somewhere on the line segment connecting A and C.
In geometry, the significance of collinear points surfaces repeatedly when establishing linearity and congruence between different figures. Evaluating potential collinearity quickly through mathematical methods like determinants simplifies geometric problem-solving, providing a clear answer without needing to physically draw or estimate.

Coordinate Geometry

Coordinate geometry, or analytical geometry, merges algebra and geometry, allowing geometric problems to be solved using algebraic equations. It’s based in a mathematical plane where points are defined by ordered pairs of numbers called coordinates.
Using coordinate geometry helps visualize and compute geometric figures, such as understanding how the formula for the area of a triangle arises from points on a plane. We utilize this by connecting algebraic expressions like determinants to spatial concepts, for instance, asserting collinearity through solutions to a determinate equation.
This approach allows transformation of complex geometric problems into algebraic problems, providing powerful analysis tools and removing ambiguity that may arise from imprecise drawings of figures.