Problem 40 - College Algebra with Modeling and Visualization

Question

Use the concept of the area of a triangle to determine if the three points are collinear. $$ (4,-5),(-2,10),(6,-10) $$

Step-by-Step Solution

Answer

The points are collinear because the area is zero.

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

The area of a triangle with vertices at points \(x_1, y_1\), \(x_2, y_2\), and \(x_3, y_3\) is calculated using the following formula: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \]If the area is zero, the points are collinear.

Step 2: Substitute the Given Points into the Formula

Substitute the coordinates \(4,-5\), \(-2,10\), and \(6,-10\) into the area formula:\[ \text{Area} = \frac{1}{2} \left| 4(10 + 10) + (-2)(-10 + 5) + 6(-5 - 10) \right| \]

Step 3: Calculate Each Term in the Equation

Perform the calculations in each step:- Calculate \ 4(10 + 10) = 80 \- Calculate \ (-2)(-10 + 5) = -2(-5) = 10 \- Calculate \ 6(-5 - 10) = 6(-15) = -90 \The area formula now is:\[ \text{Area} = \frac{1}{2} \left| 80 + 10 - 90 \right| \]

Step 4: Calculate the Absolute Value

Combine the calculated values inside the absolute value expression:\[ \left| 80 + 10 - 90 \right| = \left| 0 \right| = 0 \]

Step 5: Calculate the Final Area

Substitute the result into the area expression:\[ \text{Area} = \frac{1}{2} \times 0 = 0 \]Since the area is zero, the points are collinear.

Key Concepts

Area of a Triangle

The area of a triangle is a crucial concept in geometry, helping us understand the space enclosed by the triangle's three sides. To find the area given just the vertices of the triangle, we use a special formula based on the coordinates of these points. This is particularly useful when dealing with points on a plane in coordinate geometry. The formula used is:\[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \]Here, \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) are the coordinates of the triangle's vertices.

• The use of absolute value ensures the area is always a non-negative number.
• Multiplying by \(\frac{1}{2}\) reflects the fact that we are essentially finding the area of the parallelogram spanned by the vectors defined by these points and then halving it to get the triangle's area.

It's important to remember, if the calculated area is zero, the points lie on the same line, indicating collinearity.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, deals with defining and analyzing geometric shapes using a coordinate system. In this system:

• Points are determined by pairs of numbers known as coordinates, such as \((4, -5)\).
• These coordinates can be plotted on a plane, forming lines, shapes, and geometrical figures.

Learning coordinate geometry allows us to bring algebraic techniques into geometry. This enables the solving of geometric problems using equations and formulas. Using the given vertices, the area formula acts as a bridge between algebra (by manipulating equations) and geometry (by examining the shape).
In this case, identifying collinearity becomes more straightforward, by leveraging the algebraic expression and seeing if it simplifies to zero, thereby showing the absence of any area.

Mathematical Formulas

Mathematical formulas are essential tools that encapsulate complex relationships in a concise manner. In our problem, we employed a formula to calculate the area of a triangle with given vertices. But why are formulas important?

• Formulas provide a standard method for solving problems, offering accuracy and efficiency.
• They allow for quick computations once understood, promoting faster problem-solving skills.
• Formulas translate real-world scenarios into mathematical language, making it easier to compute and conceptualize geometric and algebraic problems.

In the context of our exercise, the outcome of the area formula tells us not only the potential size of a triangle but also about the spatial relationship of these points. When the output is zero, we conclude that the points are collinear, as no area exists among them.