Sine Rule (Part 4): Plane Vectors and Geometric Construction

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2017-10-31 16:08

This GeoGebra example demonstrates the geometric verification and dynamic construction of the sine rule in a two-dimensional plane, using rotations, segments, and circles to visually illustrate the relationship between sides and angles in a triangle. Core mathematical concepts include the sine rule, plane vectors, tangent function, and angle addition. Construction highlights: the commands Point, Segment, Rotate, Line, Angle, and Circle are used to create the triangle and its circumcircle, with sliders and text steps (Step1, Step2, Step3) guiding the step-by-step derivation. Distance and vector objects (dist, distH) assist in measurements. Suitable for high school or early college students to grasp the geometric meaning of the sine rule, and as an interactive teaching tool for instructors.

GeoGebra Commands & Parameters

Point(Circle[A, 10]) → B
Segment(A, B) → AB
Point(Circle[B, 12]) → Draw
Rotate(B, 30°, A) → B'
Line(A, B') → h
Segment(A, Point[AB, dist]) → AB'
Angle(B, A, B') → α
Segment(Hleft, Point[Segment[Hleft, Hright], distH]) → h'
Segment(A, B') → f
Segment(B, Draw) → Radius
Circle(B, 12) → Circ

AI Drawing Prompt Reference

Draw a triangle ABC with point A at the origin, point B 10 units to the right of A, and point B' obtained by rotating B 30 degrees clockwise around A. Draw line segment AB', line segment BB', and measure angle BAB' as α. Draw a circle with center B and radius 12, and draw line segment BB' from B to a point on the circle. Measure the length of line segment AB' as f, the length of line segment BB' as Radius, and the distance between the intersection points Hleft and Hright of line segment AB' and the circle as distH. Draw line segment h' starting from Hleft along the direction of h'.

The prompt above was compiled from the original drawing commands and is for reference only.