Question

In Exercises 31-32, find the area of the quadrilateral with the given vertices. (Hint: break the quadrilateral into 2 triangles.) Vertices: (0,0),(1,2),(3,0) and (4,3) .

Short answer

The area of the quadrilateral is 7 square units.

Step-by-step solution

Identify Vertices for Two Triangles

The quadrilateral can be divided into two triangles. Let's consider two triangles: Triangle 1 with vertices (0,0), (1,2), and (3,0) and Triangle 2 with vertices (1,2), (3,0), and (4,3).

Calculate Area of Triangle 1

To calculate the area of a triangle given its vertices, use the formula: \[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \]. For Triangle 1, plug in (0,0), (1,2), (3,0): \[ \text{Area}_1 = \frac{1}{2} \left| 0(2-0) + 1(0-0) + 3(0-2) \right| = \frac{1}{2} \left| -6 \right| = 3. \]

Calculate Area of Triangle 2

Using the same formula for Triangle 2 with vertices (1,2), (3,0), and (4,3): \[ \text{Area}_2 = \frac{1}{2} \left| 1(0-3) + 3(3-2) + 4(2-0) \right| = \frac{1}{2} \left| -3 + 3 + 8 \right| = \frac{1}{2} \left| 8 \right| = 4. \]

Sum the Areas of Two Triangles

The area of the quadrilateral is the sum of the areas of the two triangles: \[ \text{Total Area} = \text{Area}_1 + \text{Area}_2 = 3 + 4 = 7. \]

Key concepts

Understanding the Area of a Quadrilateral

To find the area of a quadrilateral, we can simplify our task by dividing the shape into two triangles. This is a commonly used method in geometry, especially when dealing with complex shapes. By breaking it into triangles, we use known formulas to find the area and sum them up to get the total area of the quadrilateral. This method is efficient and plays a crucial role when the quadrilateral has vertices with coordinates that make direct calculation cumbersome.

The "hint" given in this task is about leveraging this concept. By transforming a quadrilateral into simpler, calculable shapes (like triangles), we streamline the process. Understanding this approach opens up new pathways to solving similar problems across diverse geometric scenarios.

Identifying Quadrilateral Vertices

Vertices are the corner points of a shape where two sides meet. For a quadrilateral, there are four such points. In this exercise, the vertices provided are (0,0), (1,2), (3,0), and (4,3). These points lie on a coordinate plane, which is a two-dimensional space used to represent the geometric figures.

The task involves breaking the quadrilateral into two triangles by choosing different combinations of these vertices. When identifying these points:

• Ensure all points are used.
• Make sure that the triangles formed actually combine to reform the original quadrilateral.
• Each triangle uses three distinct vertices.

This setup provides the foundation for the area calculations that follow.

Triangles within Geometry

Triangles are the simplest polygons, having three sides and three angles. In geometry, they are fundamental shapes often used to simplify more complex figures like quadrilaterals.

When you look at the quadrilateral in the exercise, it becomes evident that two triangles can be strategically extracted. The first triangle uses the vertices (0,0), (1,2), and (3,0), and the second uses (1,2), (3,0), and (4,3).

Why are triangles so important? They possess straightforward area calculation formulas, and every polygon can be decomposed into triangles. This decomposition helps simplify complex area calculations. Understanding triangles is a vital skill in geometry.

Mastering the Area Calculation Formula

The area of a triangle with vertex coordinates \(x_1, y_1\), \(x_2, y_2\), \(x_3, y_3\) on a plane is calculated using the following formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \] This formula is indispensable because it allows for precise area calculations based solely on vertex coordinates without the need for additional measurements.

In the exercise solution, we computed the area for two triangles:

• For the first triangle, the vertices were plugged into the formula to get an area of 3 square units.
• For the second triangle, inserting the respective vertices into the formula gave an area of 4 square units.

Finally, the total area of the quadrilateral was found by summing these two areas, leading to an area of 7 square units. This method is not only accurate but efficient for handling such cases.