6.1: Analyzing Heron’s Formula
This activity is intended to supplement Trigonometry, Chapter 5, Lesson 2.
Time Required: 20 minutes
Activity Overview
In this activity, students will use their graphing calculators to determine the relationship between Heron’s Formula and the basic area formula.
Topics Covered
 Finding the area of a triangle
 Points of intersection
 Interpreting a graph
Teacher Preparation and Notes
 Make sure students have cleared
Y= menu before starting.  You may need to remind students how to TRACE and find points of intersection.
Associated Materials
 Student Worksheet: Analyzing Heron's Formula http://www.ck12.org/flexr/chapter/9703
Problem 1: The 3, 4, 5 right triangle
 Students should know this is a right triangle, with hypotenuse 5. Make sure that they have that the legs are 3 and 4.
 The area of this triangle is 6.
 With Heron’s Formula,
A=s(s−a)(s−b)(s−c)−−−−−−−−−−−−−−−−−√ , andY1=x(x−3)(x−4)(x−5)−−−−−−−−−−−−−−−−−−√ students might get confused with the parenthesis. Make sure all students change their window to the dimensions to the right before graphing.
WINDOW
 The graph is to the right. Have students analyze the domain and range and why there are blank spaces in the graph. The domain is
(∞,00] , [3, 4], and[5,∞) and the range is all real numbers greater than zero. If you have students zoom in further, they will see that there are nox− intercepts and oney intercept at (0, 0).

Y2=6 represents the area of this triangle. The horizontal line crosses the graph at (0.435, 6) and (6, 6). The first point, however is invalid becausex cannot be negative here. Ask students why. Explain thatx is actuallys and thats cannot be negative, because by definition it is12(a+b+c) .  Specifically, the point (6, 6) represents (
s , area). So, from this system of equations, we have determined whats is such that we have the correct area, in this case, also 6. If we can find the area in more that one manner, this will always work as a way to solve fors .
Problem 2
For this triangle, the area is,
Here, students would graph
Again,