Problem 43 - University Calculus: Early Transcendentals

Question

Find the areas of the triangles whose vertices are given. $$A(-5,3), \quad B(1,-2), \quad C(6,-2)$$

Step-by-Step Solution

Answer

The area of the triangle is 12.5 square units.

Step 1: Area Formula for Triangles

To find the area of a triangle with vertices at \((x_1, y_1),\) \((x_2, y_2),\) and \((x_3, y_3),\), we use the formula:\[Area = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|.\]

Step 2: Substitute the Coordinates

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

Step 3: Simplify the Expression

Simplify the expression step by step:\[= \frac{1}{2} \left| -5(0) + 1(-5) + 6(5) \right|\]\[= \frac{1}{2} \left| 0 - 5 + 30 \right|\]\[= \frac{1}{2} \left| 25 \right|\]

Step 4: Calculate the Area

Now, compute the area of the triangle:\[= \frac{1}{2} \times 25 = 12.5\]

Step 5: Verify the Calculation

Double-check the computations and ensure all signs and operations adhere to the formula to confirm correctness.

Key Concepts

Coordinate Geometry and Its Role in Mathematics

Coordinate geometry is a branch of mathematics that deals with the study of geometric figures using a coordinate system. It combines algebra and geometry to solve problems related to distances, angles, and areas.

• Coordinates help define the location of points precisely in a plane.
• By using this system, geometric shapes like triangles can be analyzed mathematically.

Understanding how coordinate planes work, with their x-axis and y-axis, sets the foundation for solving problems involving distances and areas in higher dimensions.
This approach simplifies calculations and leads to precise solutions in various fields, from physics to computer graphics.

Understanding the Area Formula for Triangles

The area formula for a triangle in coordinate geometry allows us to calculate the space enclosed by three points or vertices.
This formula is expressed as:
\[Area = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|,\]
where

• (x_1, y_1), (x_2, y_2), and (x_3, y_3) are the coordinates of the triangle's vertices.

This formula derives from the determinant of a matrix and provides a straightforward method to compute the area by leveraging basic arithmetic.
By applying this formula, errors in calculation are minimized, and it becomes easier to handle non-standard triangles, such as those with slanted sides.

Vertices of a Triangle: A Closer Look

Vertices are the points where two sides of a triangle meet. In coordinate geometry, vertices are represented by coordinate pairs, such as in the triangle with vertices \(A(-5, 3), B(1, -2), \) and \(C(6, -2).\)

• The location of these vertices provides essential information about the triangle's shape and size.
• Using vertices coordinates, we can apply formulas to find various properties, such as perimeter and area.

Understanding vertices helps in visualizing a triangle on the Cartesian plane, whereby the x and y coordinates reveal how each point relates to the rest.
Since each vertex has two coordinates, the data can be plotted on a graph for a clearer understanding of the triangle's orientation.

Mathematics Education: Building Strong Foundations

Mathematics education focuses on equipping students with the problem-solving skills needed to understand complex concepts.
Approaches such as teaching coordinate geometry and area calculations are designed to help students think analytically and improve their logical reasoning.

• Interactive tasks, like calculating the area of a triangle using vertices, strengthen these competencies.
• Incorporating real-world examples makes mathematical principles more relatable for learners.

This holistic approach to geometry not only improves mathematical literacy but also applies across sciences, technology, and everyday problem solving.
Emphasizing step-by-step methodologies enables students to comprehend challenging topics better and build confidence in their mathematical abilities.