Problem 64 - Algebra and Trigonometry

Question

Collinear Points and Determinants (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) Not collinear; (ii) Not collinear.

Step 1: Understand the Concept of Collinear Points and Area

If three points are collinear, they lie on the same straight line, meaning the 'triangle' they form is degenerate. Therefore, the area of such a triangle is zero.

Step 2: Apply the Determinant for Area

The area of a triangle with vertices \( (a_1, b_1), (a_2, b_2), (a_3, b_3) \) can be calculated using the formula: \[\frac{1}{2} \left| \begin{array}{ccc} a_1 & b_1 & 1 \ a_2 & b_2 & 1 \ a_3 & b_3 & 1 \end{array} \right|\] If these points are collinear, this area must be 0, which implies: \( \left| \text{the determinant} \right| = 0 \).

Step 3: Check Collinearity of Points (i)

For the points \((-6,4), (2,10), (6,13)\), calculate the determinant: \[|A| = \left| \begin{array}{ccc} -6 & 4 & 1 \ 2 & 10 & 1 \ 6 & 13 & 1 \end{array} \right|\] Calculate: \|-6(10-13) + 4(2-6) + 1(26-20)| = |-18 - 16 + 6| = |-28|\ Since this is not zero, the points are not collinear.

Step 4: Check Collinearity of Points (ii)

For the points \((-5,10), (2,6), (15,-2)\), calculate the determinant: \[|B| = \left| \begin{array}{ccc} -5 & 10 & 1 \ 2 & 6 & 1 \ 15 & -2 & 1 \end{array} \right|\] Calculate: \|-5(6 - (-2)) + 10(2 - 15) + 1(-10 - 12)| = |-5(8) + 10(-13) - 22| = |-40 - 130 - 22| = |-192|\ Since this is not zero, the points are not collinear.

Step 5: Graph to Verify Results

Graph each set of points on a coordinate plane to visually confirm they do not lie on a single straight line.

Key Concepts

Determinants

Determinants are a mathematical tool used to determine specific properties of matrices. In the context of coordinate geometry, determinants are particularly useful for finding the area of a triangle formed by three points. The formula to calculate this area using determinants is:

• For three points \( (a_1, b_1), (a_2, b_2), (a_3, b_3) \), we construct a 3x3 matrix.
• The matrix is structured as follows:
• \(\begin{array}{ccc} a_1 & b_1 & 1 \ a_2 & b_2 & 1 \ a_3 & b_3 & 1 \end{array} \)

The determinant of this matrix, when divided by two, gives us the area of the triangle. The outcome is modulated by the absolute value to ensure the area is positive. If the determinant equals zero, it means the points are collinear, meaning they lie on a single straight line and do not form a triangle with an area.

Triangle Area

The concept of finding the area of a triangle using coordinates is a practical application in coordinate geometry. It leverages the determinant formula discussed earlier. Here's how it works:

• Given three points, you can calculate the area by determining the determinant of the related matrix and dividing by 2.
• Mathematically, this is represented as:
• \(\text{Area} = \frac{1}{2} \left| \begin{array}{ccc}a_1 & b_1 & 1 \ a_2 & b_2 & 1 \ a_3 & b_3 & 1\end{array} \right| \)
• This formula simplifies the calculation of area without the need for measuring side lengths or angles.

If the area is calculated to be zero, it indicates the points are collinear. This is useful for quickly checking collinearity in plane geometry.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is a method of representing and solving geometric problems with coordinates and algebra. It allows us to use algebraic equations to describe geometric shapes and properties. In the exercise related to collinear points:

• Each point has a pair of coordinates \( (x, y) \).
• These coordinate pairs are plugged into the determinant formula to compute aspects like the area of a triangle.
• Coordinate geometry makes it easy to visualize and analyze geometrical problems using a graph.

This approach is efficient in solving not just problems of collinearity, but also in tasks such as finding the distance between points or the slope of a line through them.

Graphical Verification

Graphical verification serves as a visual check for the algebraic solutions found using determinants. After determining whether points are collinear using determinants, plotting these points on a graph verifies these results visually. Here's how it can be done:

• Plot each set of points on a Cartesian plane.
• Observe if the points are lying on a straight line or form a triangle.

In our exercise, graphing \( (-6,4), (2,10), (6,13) \) and \( (-5,10), (2,6), (15,-2) \) allowed us to visually confirm that instead of being collinear, these points form triangles. Thus, graphical verification helps bridge the gap between numerical results and intuitive understanding of geometrical concepts.