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9.1: The Pythagorean Theorem

Difficulty Level: At Grade Created by: CK-12
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This activity is intended to supplement Geometry, Chapter 8, Lesson 1.

Problem 1 – Squares on Sides Proof

1. Why is the constructed quadrilateral a square?

2. Record three sets of area measurements you made by dragging points \begin{align*}A\end{align*}, \begin{align*}B\end{align*}, and/or \begin{align*}C\end{align*}.

Square on \begin{align*}\overline{BC}\end{align*} Square on \begin{align*}\overline{AC}\end{align*} Square on \begin{align*}\overline{AB}\end{align*} Sum of squares

3. What conjecture can you make about the areas of the three squares? Does this relationship always hold when a vertex of \begin{align*}\triangle{ABC}\end{align*} is dragged to a different location?

Problem 2 – Inside a Square Proof

4. Prove that constructed quadrilateral \begin{align*}EFGH\end{align*} is a square.

5. \begin{align*}ABCD\end{align*} is a square with all sides of length \begin{align*}(x + y)\end{align*}.

The area of the square \begin{align*}ABCD\end{align*} is \begin{align*}(x + y)^2 = x^2 + 2xy + y^2\end{align*}

Each of the triangles, \begin{align*}\triangle{EFA}\end{align*}, \begin{align*}\triangle{FGD}\end{align*}, \begin{align*}\triangle{GHC}\end{align*} and \begin{align*}\triangle{HEB}\end{align*}, is a right triangle with height \begin{align*}x\end{align*} and base \begin{align*}y\end{align*}. So, the area of each triangle is \begin{align*}\frac{1}{2}xy\end{align*}.

\begin{align*}EFGH\end{align*} is a square with sides of length \begin{align*}z\end{align*}. So the area of \begin{align*}EFGH\end{align*} is \begin{align*}z^2\end{align*}.

Looking at the areas in the diagram we can conclude that:

\begin{align*}ABCD = \triangle{EFA} + \triangle{FGD} + \triangle{GHC} + \triangle{HEB} + EFGH\end{align*}

Substitute the area expressions (with variables \begin{align*}x, \ y\end{align*}, and \begin{align*}z\end{align*}) into the equation above and simplify.

6. Record three sets of numeric values for \begin{align*}\triangle{HEB}\end{align*}.

\begin{align*}BE\end{align*} \begin{align*}BE^2\end{align*} \begin{align*}HB\end{align*} \begin{align*}HB^2\end{align*} \begin{align*}BE^2 + HB^2\end{align*} \begin{align*}EH\end{align*} \begin{align*}EH^2\end{align*}

7. Does \begin{align*}BE^2 + HB^2 = EH^2\end{align*} when \begin{align*}E\end{align*} is dragged to a different locations?

8. Does \begin{align*}BE^2 + HB^2 = EH^2\end{align*} when \begin{align*}A\end{align*} or \begin{align*}B\end{align*} are dragged to different locations?

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Date Created:
Feb 23, 2012
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