Problem 56 - Algebra and Trigonometry

Question

Sketch the triangle with the given vertices, and use a determinant to find its area. $$ (1,0),(3,5),(-2,2) $$

Step-by-Step Solution

Answer

The area of the triangle is 9.5 square units.

Step 1: Understand the Vertices

The vertices of the triangle are given as \((1,0)\), \((3,5)\), and \((-2,2)\). These points represent the corners of the triangle on the Cartesian plane.

Step 2: Plotting the Triangle

To visualize the problem, plot the points \((1,0)\), \((3,5)\), and \((-2,2)\) on the Cartesian coordinate system. Draw straight lines connecting these points to outline the triangle. This helps us understand the shape we're calculating the area for.

Step 3: Use the Determinant Formula for Area

The area \(A\) of a triangle with vertices at \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\) can be found using the formula:\[A = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|\]

Step 4: Substitute the Given Points into the Formula

Using the vertices \((1,0)\), \((3,5)\), and \((-2,2)\), substitute \(x_1 = 1, y_1 = 0, x_2 = 3, y_2 = 5, x_3 = -2, y_3 = 2\) into the formula:\[A = \frac{1}{2} \left| 1(5-2) + 3(2-0) + (-2)(0-5) \right|\]

Step 5: Calculate the Determinant Value

Compute the expression inside the absolute value:\[1(5-2) + 3(2-0) + (-2)(0-5) = 1\times3 + 3\times2 + (-2)\times(-5)\]\[= 3 + 6 + 10 = 19\]

Step 6: Calculate the Triangle's Area

Apply the absolute value and multiply by \(\frac{1}{2}\):\[A = \frac{1}{2} \times 19 = 9.5\]

Step 7: Conclusion

The area of the triangle formed by the points \((1,0)\), \((3,5)\), \((-2,2)\) is \(9.5\) square units.

Key Concepts

Determinant Method

Finding the area of a triangle using the determinant method is both simple and elegant. This method hinges on the concept of calculating a determinate from a matrix formed by the coordinates of the triangle's vertices. To use this method:

• Identify the coordinates of the three vertices, which we denote as ((x_1, y_1), (x_2, y_2), (x_3, y_3)).
• Apply the determinant formula for the area as:
  \[A = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|\]
• The matrix determinant gives the application of cross-multiplication in a streamlined fashion.

With the determinant formula, the calculation is straightforward, leading to an accurate and efficient area determination.

Using it avoids common slip-ups that might occur with other geometric calculation methods.

Cartesian Plane

The Cartesian plane is the stage where our geometry unfolds. This two-dimensional plane is defined by a horizontal axis (x-axis) and a vertical axis (y-axis).

• Any point is represented by a pair of coordinates, written as (x, y).
• The plane allows for precise plotting of geometric figures, like triangles, using these points.

Visualizing a triangle entails plotting its vertices on this plane. Doing so helps students grasp the physical space the triangle occupies.

Once the points are plotted, you can simply connect them with straight lines to visualize the triangle's shape.

• Understanding the placement of each vertex is essential in accurate geometric drawings.
• It also aids in cross-referencing when using numerical methods, like the determinant method, to calculate areas or other properties.

Vertices of a Triangle

Vertices are the building blocks of a triangle, representing the triangle's corners on the plane. Each vertex is described by its coordinates:

• A vertex (x_1, y_1) marks a distinct corner.
• The set of three points ((x_1, y_1), (x_2, y_2), (x_3, y_3)) uniquely defines the triangle.

Using these vertices to calculate the triangle's area simplifies many complex computations.

• Switching between different mathematical approaches becomes easier once the vertices are identified.
• This approach allows for consistent application across different types of triangles.

Recognizing how each vertex contributes to the triangle's overall shape also helps in visualizing the geometric structure.

Geometric Visualization

Geometric visualization entails the mental image or sketching of geometric shapes – like triangles – based on given data. This skill combines both visual and analytical thinking.

• Drawing the triangle on a grid or paper based on given coordinates is a powerful tool for insight.
• It transforms abstract data into tangible representations.

Understanding the layout of a triangle before calculations provides context and can reveal relationships between its elements.

• Seeing how the triangle sits within the plane can offer clues about symmetry, dimensions, and relative angles.
• Visual aids can significantly simplify understanding mathematical and spatial concepts.

When approached effectively, visualization can make solving triangular area problems easier and lead to improved problem-solving skills.