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6.1: Analyzing Heron’s Formula

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Trigonometry, Chapter 5, Lesson 2.

Problem 1 - Consider a 3, 4, 5 right triangle.

  • Draw the triangle.
  • Find the area using \begin{align*}A=\frac{1}{2}bh\end{align*}A=12bh.
  • Consider Heron’s Formula, \begin{align*}A=\sqrt{s(s-a)(s-b)(s-c)}\end{align*}A=s(sa)(sb)(sc)

In \begin{align*}Y=\end{align*}Y=, plug into \begin{align*}Y1= \sqrt{x(x-3)(x-4)(x-5)}\end{align*}Y1=x(x3)(x4)(x5). Zoom in by changing the window. Press WINDOW and change the parameters to the right.

Press GRAPH.

WINDOW

\begin{align*}Xmin = -1\end{align*}Xmin=1

\begin{align*}Xmax = 8\end{align*}Xmax=8

\begin{align*}Xscl = 1\end{align*}Xscl=1

\begin{align*}Ymin = -1\end{align*}Ymin=1

\begin{align*}Ymax = 10\end{align*}Ymax=10

\begin{align*}Yscl = 1\end{align*}Yscl=1

\begin{align*}Xres = 1\end{align*}Xres=1

  • Describe the graph. Does it have any \begin{align*}x\end{align*}x or \begin{align*}y\end{align*}y intercepts?
  • In \begin{align*}Y2\end{align*}Y2, type \begin{align*}Y2=\frac{1}{2} \cdot 3 \cdot 4\end{align*}Y2=1234 or 6, the area of this triangle. Press GRAPH. Do the two functions intersect? If so, write the point(s) below.
  • \begin{align*}Y1\end{align*}Y1 is Heron’s formula with \begin{align*}a = 3\end{align*}a=3, \begin{align*}b = 4\end{align*}b=4, and \begin{align*}c = 5\end{align*}c=5 and \begin{align*}s = x\end{align*}s=x. What do the point(s) above tell us about this specific Heron’s formula? What do \begin{align*}(x, \ y)\end{align*}(x, y) represent?

Problem 2

Repeat the steps from Problem 1 with the triangle below.

Are your findings the same?

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Date Created:
Feb 23, 2012
Last Modified:
Nov 04, 2014
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TI.MAT.ENG.SE.1.Trigonometry.6.1