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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, 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?

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Date Created:

Feb 23, 2012

Last Modified:

Nov 03, 2014
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