Discovering the Sine Rule in Triangles
Explore the Sine Rule Discovery by measuring triangle ratios, constructing circumscribed circles, and proving the rule's validity. Understand how the ratios remain consistent across different triangles and how symmetry plays a role in the proof.
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Presentation Transcript
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings?
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . 14.5cm = . 14.5cm = . 14.5cm Construct the circle that goes through the three vertices. Measure its diameter: _____ 14.5cm Can you justify your findings?
Sine Rule Discovery Summarising: Each ratio should be the same within the limits of experimental error. Compare your results with your neighbour you have different triangles, how can the ratios be the same? What about the diameter of the circumscribed circle? Again, you should all find it is the same Let s look at how to prove our results.
Proof of the Sine Rule ? ? ? ? ? ?
Proof of the Sine Rule ? ? ? ? ? ???? =? ???? =? ? ? ? ? ? ? ? = ????? ? = ????? ????= ????= ???? ????? = ????? (from symmetry)
Proof of the Sine Rule ? ? ? ? ? ???? =? ???? =? ? ? ? ? ? ? ? = ????? ? = ????? ????= ????= ???? ????? = ????? (from symmetry)
Proof of the Sine Rule ? ? ? ? ? ?
Proof of the Sine Rule ? ? ? ? ? ?
Proof of the Sine Rule ? ? ? ? ? ? ? ? ?
Proof of the Sine Rule ? ? ? ? ? ? ? ?
Proof of the Sine Rule ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ???? = = ?? ? ????= ?? ? ? ? ????= ????= ????= ??
Proof of the Sine Rule ? ? ? ? ? 2 ? ? ? ? ? ? ? ? ???? = = ?? ? ????= ?? ? ? ? ????= ????= ????= ??
Note to Teacher To start, it might be worth asking the pupils to draw their own triangle on the reverse of the worksheet and calculate the ratios. Then they work on the given triangle. This might make the impact of getting the same ratio for everyone a lot more forceful.
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
Sine Rule Discovery Measure the side lengths and angles of the triangle as accurately as you can. Calculate the ratio ????? ?? ???? sin(????? ????????) for each side: = . = . = . Construct the circle that goes through the three vertices. Measure its diameter: _____ Can you justify your findings? SIC_2
