The Case of the Diminishing Square GXWeb

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The Case of the Diminishing Square
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Download a Geometry Expressions Model for the Diminishing Square
GXWeb Diminishing Square Model (drag point E and explore!)
Symbolic computations on this page use Nerdamer Symbolic JavaScript to complement the in-built CAS of GXWeb

A small square EFGH is constructed within a larger (unit) square, OABC, as shown.
The point P gives the locus of the defining point X(x, 0) and the AREA of the smaller square.
Can you find an algebraic model which best describes this relationship?
Drag the point X(x, 0) to explore this problem.
Try building your own model for this problem using GXweb and use it to explore both the geometry and the algebra involved.

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\(x\)
	0	0.50	1


\(f(x)\)

\(g(x)\)

\(h(x)\)

App generated by GXWeb

When x = 0.5 the area of the small square is 0.20 square units;

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1. Drag the point X(x,0) to find the area of the small square when x = 0.5.

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2. Now find the value of x that gives a small square half the area of the large square.

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3. Study the triangle OCJ. The base is 1 but can you see that the height CJ = x units, making it easy to calculate ocj(x), the AREA of the triangle in terms of x? ocj(x) =

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4. We can also let OJ be the base, and GC the perpendicular height (call it 'h'). Give an expression for the length of OJ in terms of x: oj(x) =

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5. Now calculate the area of OCJ another way using the height h and base oj(x): ocj2(x) =

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6. Use these two expressions for the same area to find height(x), height in terms of x: height(x) =

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7. Focus now on triangle OCG. can you see the same congruent triangle repeated four times around the square? If we can find the area of this triangle, then it becomes easy to find the area of the enclosed square. To find the area of triangle OCG we know the height 'h' in terms of x, so we need the base OG. This is the length OJ - JG. First, give JG. jg(x) =

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8. Now give an expression for the base (OG) in terms of x. base(x) =

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9. The area of triangle OGC is area(x) =

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10. Find an expression for the area of square EFGH - you might like to check your result using the graphical model above first! dimsquare(x) =

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How difficult did you find this activity? Please assign a rating value from 1 (very easy) to 5 (extremely difficult). Comments? Suggestions? What did you learn from this activity?

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The Case of the Diminishing Square

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