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 AC¯.

Part B
Let the point of intersection of AC¯ 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 AC¯ through point B.
Label the intersection of the line perpendicular to AC¯ through B and AC↔ point D.
Measure and display the slopes of AC↔ and BD↔.
Display the equations of AC↔ and BD↔ in the Algebra margin.
Measure and display the lengths of AC¯ and BD¯.
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?