Question

The following are the steps involved in finding the area of the triangle with vertices \((2,3),(4,7)\) and \((8,5)\). Arrange them in the sequential order from first to last. (a) \(\frac{1}{2}|(-2) 2-(-4)(-4)|=\frac{1}{2}|-20|=10\) sq. units (b) Area of triangle \(=\frac{1}{2}\left|\begin{array}{ll}\mathrm{x}_{1}-\mathrm{x}_{2} & \mathrm{x}_{2}-\mathrm{x}_{3} \\ \mathrm{y}_{1}-\mathrm{y}_{2} & \mathrm{y}_{2}-\mathrm{y}_{3}\end{array}\right|\) (c) Let \(\left(x_{1}, y_{1}\right)=(2,3),\left(x_{2}, y_{2}\right)=(4,7)\) and \(\left(x_{3}, y_{y}\right)=(8,5)\) (d) Area of triangle \(=\frac{1}{2}\left|\begin{array}{ll}2-4 & 4-8 \\ 3-7 & 7-5\end{array}\right|\) (1) abcd (2) cbda (3) \(\mathrm{cbad}\) (4) None of these

Short answer

Question: Arrange the steps in the correct order for finding the area of the triangle with vertices (2, 3), (4, 7), and (8, 5) using provided formula: Answer: (3) cbad

Step-by-step solution

Identify Vertices of the Triangle

We are given the vertices of a triangle as \((x_1,y_1) = (2,3)\), \((x_2,y_2) = (4,7)\), and \((x_3,y_3) = (8,5)\). So, the correct step is (c).

Apply the Area Formula

Now, we can find the area of the triangle using the formula: Area of triangle \(=\frac{1}{2}\left|\begin{array}{ll}x_{1}-x_{2} & x_{2}-x_{3} \\ y_{1}-y_{2} & y_{2}-y_{3}\end{array}\right|\). This is given in the step (b).

Plug in the Coordinates

Next, we need to plug in the coordinates of the vertices in the formula: Area of triangle \(=\frac{1}{2}\left|\begin{array}{ll}2-4 & 4-8 \\ 3-7 & 7-5\end{array}\right|\). This step is given as (d).

Compute the Area

Finally, we can compute the area of the triangle: \(\frac{1}{2}|(-2) 2-(-4)(-4)|=\frac{1}{2}|-20|=10\) sq. units. This corresponds to step (a). Thus, the correct sequential order is (3) \(\mathrm{cbad}\).

Key concepts

Area of a Triangle

Calculating the area of a triangle is an essential skill in coordinate geometry. When you have the vertices of a triangle in a coordinate plane, there are various methods you can use to find its area. One simple way is using the formula for a triangle's area by vertices:\[\text{Area} = \frac{1}{2} | x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) |\]This formula allows you to plug in the coordinates of the triangle's vertices and calculate the area directly.
It uses a determinant-like structure, making it efficient to use especially when dealing with complex numbers. Understanding how to apply this formula is crucial, as it allows you to solve problems rapidly without necessarily drawing a diagram.

• Label your vertices clearly and consistently to avoid confusion.
• Always use parentheses when substituting values to maintain accuracy in calculations.

Determinant Method

The determinant method for calculating the area of a triangle stems from linear algebra and offers a structured approach. This method involves using the determinant of a matrix to yield the area value. Here's a structured breakdown of this process:
Given three vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\), construct a 2x2 matrix using subtraction on each row:
\[\frac{1}{2} \left| \begin{array}{cc}x_1-x_2 & x_2-x_3 \y_1-y_2 & y_2-y_3 \end{array} \right|\]
The determinant of this matrix provides you with twice the area of the triangle when you consider absolute value, and hence, it must be halved to get the actual area.

• Matrix construction is key; ensure the order of subtraction is correct.
• This method highlights the beauty of symmetry and consistency in mathematics.

It's an elegant technique often introduced in intermediate-level math courses, showcasing the interconnectedness of algebra and geometry.

Vertices of a Triangle

Vertices are the corner points of a triangle, usually represented in coordinate geometry by pairs of coordinates. In coordinate geometry, identifying vertices correctly is fundamental to calculating various properties like area, perimeter, and more.
Each triangle has three vertices, typically denoted as \((x_1, y_1), (x_2, y_2), (x_3, y_3)\). These points define the shape and size of the triangle in space.

• Specifying vertices involves marking the exact coordinates, ensuring precision.
• Properly identifying these points is the first critical step before applying any calculation-based formula such as the area.

Not only do vertices define the triangle, but they also serve as an entry point for exploring other geometric transformations, such as translations and rotations, that could be applied to the triangle within the coordinate plane.