Problem 46 - A Graphical Approach to College Algebra

Question

Use the concept of the area of a triangle discussed in Exercises \(39-44\) to determine whether the three points are collinear. $$(3,6),(-1,-6),(5,11)$$

Step-by-Step Solution

Answer

The points are not collinear.

Step 1: Formula for Area of Triangle

To determine if the points \((3,6), (-1,-6), (5,11)\) are collinear, we use the formula for the area of a triangle given three vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\): \[A = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|\]If the area \(A\) is zero, then the points are collinear.

Step 2: Substitute the Points into the Formula

Substitute the given points \((3,6), (-1,-6), (5,11)\) into the formula:\[A = \frac{1}{2} \left| 3(-6-11) + (-1)(11-6) + 5(6+6) \right|\]

Step 3: Simplify Inside the Absolute Value

Calculate the values inside the absolute value:\[= \frac{1}{2} \left| 3(-17) + (-1)(5) + 5(12) \right|= \frac{1}{2} \left| -51 - 5 + 60 \right|= \frac{1}{2} \left| 4 \right|\]

Step 4: Calculate the Area

Now compute the area:\[A = \frac{1}{2} \times 4 = 2\]

Step 5: Check for Collinearity

Since the area \(A = 2\) is not zero, the points \((3,6), (-1,-6), (5,11)\) are not collinear.

Key Concepts

Area of a Triangle

The area of a triangle is a concept from geometry used to measure the surface covered by the triangle's three sides. To find the area of a triangle when you have its vertices in a coordinate system, we use a specific formula that leverages the coordinates of its vertices.

• If you're given the three points as coordinates, say \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), the area \(A\) can be calculated using:
  
  \[A = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|\]

This formula outputs the area covered within the triangle. If the calculated area is zero, it means all three points lie on a single straight line, which implies they are collinear.

Vertices

Vertices are the corners or points where two or more lines meet; in the context of geometry, especially triangles, they are the endpoints of the triangle’s sides. For a triangle denoted by three points \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), each of these points represents a vertex.
These vertices determine the shape and size of a triangle.

• In practical problems, knowing the coordinates of vertices allows you to compute other properties, such as side lengths and angles, using the distance formula or trigonometric ratios.
• When vertices of a triangle are given in a coordinate plane, one can also explore properties such as collinearity for real-world applications.

Absolute Value

Absolute value denotes the magnitude or size of a number, regardless of its sign. It tells us how far a number is from zero on the number line.
In the context of the area of a triangle, we apply it to ensure that the area calculation remains non-negative.

• For example: If computing the area, and an expression yields a negative result, applying the absolute value \( |x| \) converts it to positive.
• This is crucial when you calculate algebraic expressions, especially in determinants or geometric computations, to avoid negative interpretations of physical quantities like area.
• The formula for the area of the triangle deliberately includes absolute value to accommodate any order of vertex insertion as it does not impact the area measurement.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where we use a system of coordinates to define geometric elements and relations.
It allows us to address problems that involve finding distances, midpoints, gradients, and angles using algebra.

• Using Cartesian coordinates, any geometric shape or line can be described with algebraic equations.
• It bridges geometry and algebra, allowing the use of graphing and equations to solve geometric problems like areas or collinear checks, as shown with triangles here.
• In our example, using coordinate geometry, we plug in vertex coordinates directly into a formula to reveal if points align on a straight line, i.e., if they are collinear.