Problem 30 - APEX Calculus

Question

Find the area of the triangle with the given vertices. Vertices: (3,1),(1,2) and (4,3) .

Step-by-Step Solution

Answer

The area of the triangle is 2.5 square units.

Step 1: Understand the Problem

We need to find the area of a triangle given its vertices: \((3,1)\), \((1,2)\), and \((4,3)\). We will use the formula for the area of a triangle with coordinates \((x_1, y_1), (x_2, y_2), (x_3, y_3)\).

Step 2: Apply the Area Formula

The area \(A\) of a triangle with vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) is given by:\[A = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|\]Substituting the coordinates, we get:\[A = \frac{1}{2} \left| 3(2-3) + 1(3-1) + 4(1-2) \right|\]

Step 3: Calculate Each Term

Calculate each term of the expression:- First term: \(3(2-3) = 3(-1) = -3\).- Second term: \(1(3-1) = 1(2) = 2\).- Third term: \(4(1-2) = 4(-1) = -4\).

Step 4: Compute the Expression

Add the results of the terms within the absolute value:\[A = \frac{1}{2} \left| -3 + 2 - 4 \right| = \frac{1}{2} \left| -5 \right|\]

Step 5: Calculate the Area

Find the absolute value and divide by 2:\[A = \frac{1}{2} imes 5 = 2.5\]

Key Concepts

Triangle Area Formula

The area of a triangle can be calculated using a special formula if you know the coordinates of its vertices. This is a useful technique in coordinate geometry and simplifies the process greatly. When the vertices are expressed as coordinates

• (x_1, y_1),
• (x_2, y_2),
• and (x_3, y_3),

the area formula is:\[A = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|\]This formula works by computing a determinant of a matrix derived by the vertex coordinates. It effectively breaks down the area computation into a sequence of simple arithmetic operations. Using this formula, you can quickly calculate the area without needing to plot the triangle or use more complex geometry principles.Make sure to treat the value inside the formula carefully, as minor arithmetic or sign errors could lead to incorrect answers. Additionally, keep in mind that this formula returns the absolute area value, ensuring it is always positive. After you understand the formula, it becomes a reliable tool in solving geometry problems in the coordinate plane.

Vertex Coordinates

Vertex coordinates are essential when it comes to solving geometry problems involving shapes like triangles in the coordinate system. Each vertex of a triangle is represented by an

• ordered pair (x, y), which locates the point on the Cartesian plane.

The example provided involves the vertices

• (3, 1),
• (1, 2),
• and (4, 3).

Understanding these coordinates is crucial since they are used in the triangle area formula to calculate the exact area under discussion. When using the coordinates, align each point carefully according to the axis. This practice helps avoid errors when substituting values into formulas. Consider how each coordinate might change or interact with others during computations, like determining distances or using formulas for geometric properties. Practicing these steps ensures accuracy and deepens comprehension of coordinate-based problem-solving.

Absolute Value

Absolute value is a mathematical concept that often comes into play when dealing with real numbers and computations in algebra and geometry, including area formulas. It measures the distance a number is from zero on the number line. For example, both -5 and 5 have an absolute value of 5, denoted as \[| -5 | = | 5 | = 5.\]In the context of calculating the area of a triangle using coordinates, the area formula involves taking the absolute value of an expression. This operation ensures that the result always reflects the true magnitude of the area, without regard to its original mathematical sign. This is because an area should never be a negative figure; it is always a positive measure.In practical terms, when you compute\[\left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|\]it's necessary to use absolute value to interpret the expression correctly before dividing by two. This step is pivotal in getting accurate results no matter how the values of x and y from the vertices alter the interim result of the operation.