Problem 48 - 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. $$(4,-5),(-2,10),(6,-10)$$

Step-by-Step Solution

Answer

The points (4,-5), (-2,10), and (6,-10) are collinear.

Step 1: Recall the Formula for the Area of a Triangle

The formula for the area of a triangle given by three points \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) is \((1/2) | x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) |\). This formula calculates the area of a triangle formed by the given vertices.

Step 2: Substitute the Given Points into the Formula

Substitute the points \( (4, -5), (-2, 10), (6, -10) \) into the formula:\[(1/2) | 4(10 - (-10)) + (-2)((-10) - (-5)) + 6((-5) - (10)) |.\]

Step 3: Perform the Arithmetic Operations Inside the Formula

Calculate each part inside the formula:\[ 4(10 + 10) = 4 \times 20 = 80 \,\ (-2)(-10 + 5) = -2 \times (-5) = 10 \,\ 6(-5 - 10) = 6 \times (-15) = -90. \]

Step 4: Calculate the Expression Inside the Absolute Value

Combine the values obtained:\[ 80 + 10 - 90 = 0. \]

Step 5: Apply the Absolute Value and Multiply by 1/2

Since the expression inside the absolute value is 0, we have:\[(1/2) | 0 | = 0. \]

Step 6: Interpret the Result

If the area is 0, it means the given points are collinear, lying on a straight line.

Key Concepts

Area of a Triangle

The area of a triangle is a measure of the region enclosed by the three sides of the triangle. In coordinate geometry, when you have three points, you can use a formula to find the area formed by these points. If the area is zero, the points lie on a straight line, i.e., they are collinear. The formula used to calculate the area of a triangle given three points \(x_1, y_1\), \(x_2, y_2\), and \(x_3, y_3\) is \[\frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|.\]Here's how it works:

• It uses the coordinates of the points to compute the determinant, which represents the "spread" of the points.
• The operations inside the formula reflect the changes in vertical distances (height dimension) scaled by the horizontal positions (base dimension).
• The absolute value ensures that the area is non-negative, and the division by 2 accounts for the triangle formed by two vectors.

Understanding this formula is essential to determining collinearity in coordinate geometry.

Coordinate Geometry

Coordinate geometry is a branch of mathematics that allows for geometrical forms to be expressed in numerical format using coordinates. In the case of a triangle, the vertices of the triangle are given as coordinate points in a plane, say \(x_1, y_1\), \(x_2, y_2\), and \(x_3, y_3\).
This method helps us find distances, midpoints, slopes, and even the area of geometric shapes using constructed formulas.

• Each point is a set of numerical values that give its precise location in the plane.
• This precision is key for calculations like finding intersections or determining collinearity.
• The concept of collinearity, derived using coordinate geometry, means that the points lie on the same straight line, which leads to an area of zero for a triangle formed by these points.

By understanding these basic principles, you can unlock powerful tools allowing precise spatial analysis and recognition of patterns in geometry.

Arithmetic Operations

Arithmetic operations form the basis of problem-solving in mathematics, especially when working with geometric calculations. In the given exercise, arithmetic plays a crucial role in entering values into the area formula correctly. Let's break down how addition, subtraction, multiplication, and absolute values function within this context:

• Addition and Subtraction: Key for combining and simplifying expressions that involve coordinate differences. They help in finding the net distance change in the \(y\)-values.
• Multiplication: Used to scale these differences by the respective \(x\)-coordinate each time.
• Absolute Values: They ensure that we're measuring the positive "spread" of points by making the area always non-negative.

Performing these operations in the right sequence ensures accurate computation of the expression determining the triangle's area.
Errors in simple arithmetic could lead to incorrect conclusions about whether points are collinear. Thus, understanding these operations is key for investigating geometrical properties with precision.