How to find the area of a triangle using vertices?
Understand the Problem
The question is asking how to calculate the area of a triangle given its vertices. This likely involves using the coordinates of the triangle's vertices in a mathematical formula.
Answer
The area of the triangle is $7.5$.
Answer for screen readers
The area of the triangle is $7.5$ square units.
Steps to Solve
- Identify the vertices' coordinates
Let's say the triangle has vertices at points $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$. For instance, we can assume:
- Point $A(1, 2)$
- Point $B(4, 5)$
- Point $C(7, 3)$
- Use the formula to calculate area
The area $A$ of the triangle can be calculated using the formula:
$$ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| $$
This formula uses the coordinates of the vertices directly.
- Substitute the coordinates into the formula
Now we substitute the coordinates of our points into the formula:
$$ A = \frac{1}{2} \left| 1(5 - 3) + 4(3 - 2) + 7(2 - 5) \right| $$
- Perform the calculations
Calculating inside the absolute value:
$$ A = \frac{1}{2} \left| 1(2) + 4(1) + 7(-3) \right| $$
This simplifies to:
$$ A = \frac{1}{2} \left| 2 + 4 - 21 \right| $$
- Complete the calculation
Now, summing the terms:
$$ A = \frac{1}{2} \left| -15 \right| = \frac{1}{2} \times 15 = 7.5 $$
The area of the triangle is $7.5$ square units.
More Information
This method of calculating the area of a triangle is derived from determinants in coordinate geometry, allowing you to find the area simply by using the coordinates of the vertices without needing to calculate distances or heights directly.
Tips
- Incorrectly substituting coordinates: Ensure that the coordinates match the vertices in proper order.
- Forgetting the absolute value: The area must always be a positive value, so be careful to include the absolute value in your calculations.
