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Tangent Line to a Circle

Tangent Line to a Circle
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2013-12-16 08:43
CircleCoordinate GeometryDerivativeGeometric Construction
This example demonstrates how to find the equation of a tangent line to a circle given the point of contact. It involves core math concepts like circle equations, tangent properties, slope calculation, and perpendicular relationships. GeoGebra commands/objects used include If, GCD, Segment, Slope, OrthogonalLine, Intersect, Midpoint, Circle, Tangent, Rotate, Angle, suitable for high school coordinate geometry and geometric construction exercises.

GeoGebra Commands & Parameters

  1. If(y(C) < 0, ")", "") → brY_AR
  2. If(y(C) < 0, "(", "") → brY_AL
  3. If(x(C) < 0, ")", "") → brX_AR
  4. If(x(C) < 0, "(", "") → brX_AL
  5. If(c / d < 0, "-", "") → sign
  6. GCD(c, d) → HCF
  7. If(c > 0 ∧ d > 0 ∧ HCF < 2 ∧ d ≠ 1, "", If[den ≟ 0, "= undefined", If[den ≠ 1, "=" + sign + "\frac{" + num + "}{" + den + "}", "=" + sign + num]]) → text11
  8. Segment(C, P) → a
  9. Slope(a) → b
  10. OrthogonalLine(C, line) → g
  11. OrthogonalLine(P, g) → h
  12. Intersect(g, h) → C_1
  13. Intersect(line, g) → D
  14. Intersect(g, h) → E
  15. Intersect(h, a) → F
  16. Intersect(line, g) → G
  17. Midpoint(C, C_1) → H
  18. Midpoint(P, C_1) → I
  19. Circle(C, P) → p
  20. Tangent(P, p) → m,
  21. Rotate(C, 90°, K) → C'
  22. Rotate(C, 90°, P) → C'_1
  23. Angle(C, P, C'_1) → β
  24. If(c / d < 0 ∨ d ≟ 0, "", "-") → sign_1

AI Drawing Prompt Reference

Draw a circle with a center point and a point on the circle as the point of tangency. Construct the tangent line at that point, then draw a line perpendicular to the tangent passing through the center. Next, draw another line through the tangency point perpendicular to that perpendicular line, intersecting the tangent. Connect the center and tangency point to form a radius, and display its slope. Mark a right angle at the tangency point to show the radius is perpendicular to the tangent. Also, rotate the center 90 degrees around the tangency point to create an auxiliary point, and display that rotation angle. Finally, show text near the figure displaying the slope of the tangent line, simplified if it is a fraction.
The prompt above was compiled from the original drawing commands and is for reference only.