Question

Area of a Triangle Finding the area of a triangle is straightforward if you know the length of the base and the height of the triangle. But is it possible to find the area of a triangle if you know only the coordinates of its vertices? In this task, you’ll find out. Consider ΔABC, whose vertices are A(2, 1), B(3, 3), and C(1, 6); let line segment AC represent the base of the triangle.

Part A Find the equation of the line passing through B and perpendicular to .
Part B Let the point of intersection of with the line you found in part A be point D. Find the coordinates of point D.
Part C Use the distance formula to find the length of the base and the height of ΔABC.
Part D Find the area of ΔABC.
Part E Now check your work by using the GeoGebra geometry tool to repeat parts A through D. Open GeoGebra, and complete each step below. If you need help, follow these instructions for using GeoGebra. You will take a screenshot of your work when you are through, so be sure to clearly label your construction as you work. Plot points A, B, and C, and draw a polygon, ΔABC, through the points. Draw a line perpendicular to through point B. Label the intersection of the line perpendicular to through B and point D. Measure and display the slopes of and . Display the equations of and in the Algebra margin. Measure and display the lengths of and . Calculate and display the area of ΔABC. Take a screenshot showing the geometric construction and the Algebra margin, save it, and insert the image in the space below.
Part F Compare the calculations displayed in GeoGebra with the calculations you completed in parts A through D. Look in the Algebra margin too. Do the results in GeoGebra match the results you obtained earlier? If not, where do the discrepancies occur? You might have to rearrange equations algebraically to determine whether two equations match.
Part G You’ve seen two methods for finding the area of ΔABC—using coordinate algebra (by hand) and using geometry software. How are the two methods similar? How are they different? Why might coordinate algebra be important in making and using geometry software?

Answer 1

To find the area of a triangle using the coordinates of its vertices, we can employ a series of algebraic and geometric methods, including deriving line equations and using distance formulas, or alternatively, utilize geometry software such as GeoGebra for visualization and calculation.

Step-by-step explanation:

To find the area of a triangle with vertices at coordinates, we can use a sequence of steps involving finding slopes, equations of lines, and distances between points. Considering ΔABC with vertices A(2, 1), B(3, 3), and C(1, 6), we must:

1. Find the equation of the line that passes through B and is perpendicular to line AC.
2. Determine the point of intersection D between this new line and line AC.
3. Use the distance formula to calculate the lengths of AC (base) and BD (height).
4. Compute the area of ΔABC using the formula 1/2 × base × height.

By following these steps or using a geometry software tool like GeoGebra, we can find the required area both algebraically and geometrically and compare the results.