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 BC¯¯¯¯¯¯¯¯
Square on AC¯¯¯¯¯¯¯¯
Square on 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 △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=x2+2xy+y2
Each of the triangles, △EFA, △FGD, △GHC and △HEB, is a right triangle with height x and base y. So, the area of each triangle is 12xy.
EFGH is a square with sides of length z. So the area of EFGH is z2.
Looking at the areas in the diagram we can conclude that:
Substitute the area expressions (with variables x, y, and z) into the equation above and simplify.
6. Record three sets of numeric values for △HEB.
7. Does BE2+HB2=EH2 when E is dragged to a different locations?
8. Does BE2+HB2=EH2 when A or B are dragged to different locations?