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

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

Problem 2 – Inside a Square Proof

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

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

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

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

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

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*}ABCD=EFA+FGD+GHC+HEB+EFGH

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

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

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

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

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

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Date Created:
Feb 23, 2012
Last Modified:
Nov 03, 2014
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