<meta http-equiv="refresh" content="1; url=/nojavascript/">
You are reading an older version of this FlexBook® textbook: CK-12 Texas Instruments Geometry Student Edition Go to the latest version.

# 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 $A$, $B$, and/or $C$.

Square on $\overline{BC}$ Square on $\overline{AC}$ Square on $\overline{AB}$ 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 $\triangle{ABC}$ is dragged to a different location?

## Problem 2 – Inside a Square Proof

4. Prove that constructed quadrilateral $EFGH$ is a square.

5. $ABCD$ is a square with all sides of length $(x + y)$.

The area of the square $ABCD$ is $(x + y)^2 = x^2 + 2xy + y^2$

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

$EFGH$ is a square with sides of length $z$. So the area of $EFGH$ is $z^2$.

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

$ABCD = \triangle{EFA} + \triangle{FGD} + \triangle{GHC} + \triangle{HEB} + EFGH$

Substitute the area expressions (with variables $x, \ y$, and $z$) into the equation above and simplify.

6. Record three sets of numeric values for $\triangle{HEB}$.

$BE$ $BE^2$ $HB$ $HB^2$ $BE^2 + HB^2$ $EH$ $EH^2$

7. Does $BE^2 + HB^2 = EH^2$ when $E$ is dragged to a different locations?

8. Does $BE^2 + HB^2 = EH^2$ when $A$ or $B$ are dragged to different locations?

Feb 23, 2012

Nov 03, 2014