Finding Triangle Area Using Vertices Coordinates

The question asks for the area of a triangle formed by joining three specific points or vertices: (3,0), (0,6), and (-5,0).

There are a couple of ways to calculate the area of a triangle when you know the coordinates of its vertices. Let's explore the common methods.

Method 1: Using the Determinant Formula

Given the three vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \), the area of the triangle can be calculated using the formula:

\( \text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \)

Let's assign the given points:

• \( (x_1, y_1) = (3, 0) \)
• \( (x_2, y_2) = (0, 6) \)
• \( (x_3, y_3) = (-5, 0) \)

Now, substitute these values into the formula:

\( \text{Area} = \frac{1}{2} |3(6 - 0) + 0(0 - 0) + (-5)(0 - 6)| \)

Perform the calculations step-by-step:

\( \text{Area} = \frac{1}{2} |3(6) + 0(0) + (-5)(-6)| \)

\( \text{Area} = \frac{1}{2} |18 + 0 + 30| \)

\( \text{Area} = \frac{1}{2} |48| \)

\( \text{Area} = \frac{1}{2} \times 48 \)

\( \text{Area} = 24 \)

So, the area of the triangle using the determinant formula is 24 square units.

Method 2: Using Base and Height

Sometimes, if the vertices lie on the axes, it's easier to use the base and height formula for a triangle: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \).

Observe the given vertices: (3,0), (0,6), and (-5,0).

Two of the vertices, (3,0) and (-5,0), lie on the x-axis (since their y-coordinates are 0). This forms a convenient base for the triangle along the x-axis.

The length of the base is the distance between these two points on the x-axis:

\( \text{Base} = |3 - (-5)| = |3 + 5| = |8| = 8 \text{ units} \)

The height of the triangle is the perpendicular distance from the third vertex (0,6) to the line containing the base (which is the x-axis in this case). The distance from a point \( (x, y) \) to the x-axis is \( |y| \). So, the height is the absolute value of the y-coordinate of (0,6).

\( \text{Height} = |6| = 6 \text{ units} \)

Now, calculate the area using the base and height:

\( \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \)

\( \text{Area} = \frac{1}{2} \times 8 \times 6 \)

\( \text{Area} = \frac{1}{2} \times 48 \)

\( \text{Area} = 24 \)

Using the base and height method also gives an area of 24 square units.

Both methods confirm that the area of the triangle with vertices (3,0), (0,6), and (-5,0) is 24 square units.

Revision Table: Area of Triangle Concepts

Additional Information: Coordinate Geometry Area

The determinant formula for the area of a triangle is derived from vector cross products or by using the Shoelace formula. It is a general method that works for any three points in the Cartesian plane, regardless of whether they lie on the axes.

Understanding how to calculate area using coordinates is fundamental in coordinate geometry and is useful for various problems, such as finding the area of polygons or determining if points are collinear (if the area of the triangle formed by three points is 0, they are collinear).

For this specific problem, the points simplify the calculation significantly when using the base and height method because two points share the same y-coordinate (0), making the segment connecting them horizontal and easily measurable on the x-axis.