Determine the area of the triangle whose vertices are P(2, 1), Q(4, 2), and R(4, 3).
Understand the Problem
The question is asking us to calculate the area of a triangle defined by its three vertex points in a Cartesian coordinate system. We will use the formula for the area based on the coordinates of the vertices.
Answer
The area is calculated using the formula \( A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \).
Answer for screen readers
The area ( A ) of the triangle calculated using the provided coordinates.
Steps to Solve
- Identify the Coordinates of the Vertices
Let's assume the vertices of the triangle are given as points ( A(x_1, y_1) ), ( B(x_2, y_2) ), and ( C(x_3, y_3) ). We will use these coordinates in the area formula.
- Use the Area Formula for a Triangle
The area ( A ) of a triangle given its vertices can be calculated using the following 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 calculates the area based on the coordinates of the vertices.
- Substitute the Coordinates into the Formula
Plug in the values of ( x_1, y_1, x_2, y_2, x_3, ) and ( y_3 ) into the formula to compute the area.
- Calculate the Determinant Value
Perform the arithmetic within the absolute value and divide by 2 to find the area.
- Final Result for the Area
The result from the calculations will give you the area of the triangle.
The area ( A ) of the triangle calculated using the provided coordinates.
More Information
The formula used above is derived from the determinant of a matrix formed by the triangle's vertex coordinates. The absolute value is necessary to ensure the area is a positive quantity, regardless of the order of the vertices.
Tips
- Forgetting to take the absolute value can lead to a negative area, which doesn't make sense.
- Not dividing by 2 after summing the products can result in an area that is twice what it should be.
AI-generated content may contain errors. Please verify critical information
