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# 9.1: The Pythagorean Theorem

Difficulty Level: At Grade Created by: CK-12

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 A\begin{align*}A\end{align*}, B\begin{align*}B\end{align*}, and/or C\begin{align*}C\end{align*}.

Square on BC¯¯¯¯¯¯¯¯\begin{align*}\overline{BC}\end{align*} Square on AC¯¯¯¯¯¯¯¯\begin{align*}\overline{AC}\end{align*} Square on AB¯¯¯¯¯¯¯¯\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 ABC\begin{align*}\triangle{ABC}\end{align*} is dragged to a different location?

## Problem 2 – Inside a Square Proof

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

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

The area of the square ABCD\begin{align*}ABCD\end{align*} is (x+y)2=x2+2xy+y2\begin{align*}(x + y)^2 = x^2 + 2xy + y^2\end{align*}

Each of the triangles, EFA\begin{align*}\triangle{EFA}\end{align*}, FGD\begin{align*}\triangle{FGD}\end{align*}, GHC\begin{align*}\triangle{GHC}\end{align*} and HEB\begin{align*}\triangle{HEB}\end{align*}, is a right triangle with height x\begin{align*}x\end{align*} and base y\begin{align*}y\end{align*}. So, the area of each triangle is 12xy\begin{align*}\frac{1}{2}xy\end{align*}.

EFGH\begin{align*}EFGH\end{align*} is a square with sides of length z\begin{align*}z\end{align*}. So the area of EFGH\begin{align*}EFGH\end{align*} is z2\begin{align*}z^2\end{align*}.

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

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

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

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

BE\begin{align*}BE\end{align*} BE2\begin{align*}BE^2\end{align*} HB\begin{align*}HB\end{align*} HB2\begin{align*}HB^2\end{align*} BE2+HB2\begin{align*}BE^2 + HB^2\end{align*} EH\begin{align*}EH\end{align*} EH2\begin{align*}EH^2\end{align*}

7. Does BE2+HB2=EH2\begin{align*}BE^2 + HB^2 = EH^2\end{align*} when E\begin{align*}E\end{align*} is dragged to a different locations?

8. Does BE2+HB2=EH2\begin{align*}BE^2 + HB^2 = EH^2\end{align*} when A\begin{align*}A\end{align*} or B\begin{align*}B\end{align*} are dragged to different locations?

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