Objective: To prove that the dots (x,y) at the corner of the carpenter's square obtained from the procedure depicted in Figures 2 - 6 lie on a circle.
Procedure: We determine the equation of a circle that contains the points (x,y).
Answer the following questions.
(a) Based on the construction procedure, what determines the length of the diameter of the circle we seek?
(b) Using Figure 7, which is repeated below for convenience, what is the radius of the circle we seek?
(c) Using Figure 7, what is the center of the circle we seek?
The answers to the preceding questions may help you recognize features of the algebraic steps that we develop below.
Note that the dashed line from vertex C is a perpendicular to segment AB. This gives us two other right triangles as shown in Figure 8; right triangles AEC and BEC.
Next observe that the length of segment EC is y. (Explain.)
Applying the Pythagorean theorem to triangle BEC we find the length of segment CB is
Now we know the length of all three sides of triangle ACB since segment AB has length d. See Figure 9.
Finally apply the Pythagorean Theorem to triangle ACB. We get
Expanding this equation, simplifying, and rearranging we obtain
Next we complete the square on the terms containing x to get
Finally we can express the first three terms as a perfect square:
This final equation represents a circle centered at (d/2, 0) with radius d/2.
Compare this result with your answers to questions (b) and (c) above.
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