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

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

Press GRAPH.

WINDOW

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

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

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

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

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

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

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

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

## Problem 2

Repeat the steps from Problem 1 with the triangle below.

Are your findings the same?

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