thales’ theorem. easily constructible right triangle draw a circle. draw a line using the...
TRANSCRIPT
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Thales’ Theorem
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Easily Constructible Right Triangle
Draw a circle.
Draw a line using the circle’s center and radius control points.
Construct the intersection of the line and circle.
Label the intersection points A and C.
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ConstructionSo Far
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Finishing the Construction
Draw a third point somewhere on the circle. Label this point B.
Connect the three points on the circle with line segments to form a triangle.
Measure ∠ABC.
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Result
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Thales’ Theorem
Thale’s Theorem: An inscribed angle in a semicircle is a right angle1.
1 Weisstein, Eric W. “Thales’ Theorem.” From Mathworld--A Wolfram Web Resource.
http://mathworld.wolfram.com/ThalesTheorem.html
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Verify
Let’s verify that this always works.
Drag point B around the circle.
Does the measurement stay at 90°?
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Create a New Document
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Application
We can use Thales’ Theorem to construct the tangent to a circle that passes through a given point2.
Start by drawing a circle and a point outside of the circle.
Label the circle’s center O and the point P.
2 Wikipedia contributors, ‘Thales’ theorem’, Wikipedia, The Free Encyclopedia,
http://en.wikipedia.org/w/index.php?title=Thales%27_theorem&oldid=417586850
(accessed March 18, 2011).
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Initial Figure
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Application (Cont.)
Draw the line segment OP.
Construct the midpoint of OP and label it H.
Draw a circle with center H and radius P.
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ConstructionSo Far
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Application (Cont.)
Construct the intersections of the circles.
Label the intersections S and T.
Draw the lines PS and PT.
Note how these lines are tangent to the original circle!
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Result
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Application (Cont.)
Measure ∠OSP and ∠OTP.
Can you see the use of Thales’ Theorem?
Where else might Thales’ Theorem be useful?
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Conclusion