Problem 53 - Algebra and Trigonometry

Question

\(51-54\) : Sketch the triangle with the given vertices and use a determinant to find its area. $$ (-1,3),(2,9),(5,-6) $$

Step-by-Step Solution

Answer

The area of the triangle is 31.5 square units.

Step 1: Plot the Triangle's Vertices

First, let's identify the vertices of the triangle, which are given as \((-1, 3)\), \((2, 9)\), and \((5, -6)\). Sketch the triangle on a coordinate plane by plotting these points and drawing lines between them to form the triangle.

Step 2: Write the Formula for the Area Using a Determinant

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

Step 3: Substitute the Vertices into the Formula

Substitute the coordinates of the vertices into the determinant formula. Let \((x_1, y_1) = (-1, 3)\), \((x_2, y_2) = (2, 9)\), and \((x_3, y_3) = (5, -6)\). The determinant becomes: \[ A = \frac{1}{2} \left| -1(9 - (-6)) + 2((-6) - 3) + 5(3 - 9) \right|.\]

Step 4: Calculate the Determinant

Calculate each part of the determinant from Step 3: \[-1(9 + 6) = -1 \times 15 = -15,\]\[2(-6 - 3) = 2 \times (-9) = -18,\]\[5(3 - 9) = 5 \times (-6) = -30.\] Add these values together: \(-15 - 18 - 30 = -63.\)

Step 5: Compute the Final Area

Insert the result from Step 4 into the final formula to get the area: \[ A = \frac{1}{2} \times |-63| = \frac{1}{2} \times 63 = 31.5.\] Thus, the area of the triangle is 31.5 square units.

Key Concepts

Vertices of a Triangle

The vertices of a triangle are simply the points where the sides of the triangle meet. Each vertex is defined by a pair of coordinates; in our example, the vertices are \((-1, 3)\), \((2, 9)\), and \((5, -6)\).
Visualizing these points on a coordinate plane helps us understand the shape and location of the triangle.
To sketch the triangle, plot each vertex and connect the dots with straight lines. This forms our triangle.

• The first vertex, \((-1, 3)\), is on the left side of the coordinate plane.
• The second vertex, \((2, 9)\), is higher up, as it has a larger \(y\) value.
• The third vertex, \((5, -6)\), is much lower, due to its negative \(y\) value.

This arrangement creates a unique triangle shape, which is necessary for further calculations.

Determinant Method

The determinant method is a mathematical technique used to find the area of a triangle when its vertices are known. This method is particularly useful with triangles on a coordinate plane.
The determinant itself is calculated from the coordinates of the triangle's vertices. It is a specific arithmetic expression that can be transformed into a simple area formula.

• This method removes the need for more common geometry methods like using base and height, making it efficient and effective for coordinate geometry.
• It involves plugging the coordinates into a formula that seems complex at first but simplifies area calculation significantly.

Understanding the determinant is crucial because it transforms complex operations into an easy-to-use formula, perfect for students learning coordinate geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric objects using a coordinate system.
It's what allows us to use algebraic methods to solve geometric problems. By using coordinates, we can locate points in space, like the vertices of our triangle.

• The coordinate plane is a two-dimensional space defined by an x-axis (horizontal) and y-axis (vertical).
• Points are defined by their \(x\) and \(y\) coordinates, such as \((-1, 3)\), which means \(-1\) on the \(x\)-axis and \(3\) on the \(y\)-axis.

This method helps in visualizing and calculating areas of geometric shapes efficiently. It's an essential tool in the study of mathematics, physics, engineering, and computer science.

Area Calculation Formula

The area calculation formula using determinants provides a straightforward way to find the area of a triangle from its vertices.
In our case, the formula is:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Here, the \(x_i\) and \(y_i\) represent the coordinates of the triangle's vertices.

• You substitute the coordinate values into the equation.
• The absolute value ensures that the area is a positive number, as a negative area doesn't make sense.
• The \(\frac{1}{2}\) is necessary since this formula derives from a determinant calculation.

Using this method provides a clear path from understanding vertex coordinates to finding the area, helping students connect algebra to geometry seamlessly.