The Lions Field

The Lion’s Field Ajani's Notebook:

Ajani only knew a few pieces of information which he had to use to solve how far he needed to walk to reach Eagle Rock. He could only go west or east because crossing the field would be too dangerous for the lamb. He also needed to know how fast he had to travel to reach Eagle Rock in two days time. To organize what he knew Ajani made a list with a corresponding drawing:

Ajani’s List: The Eastern Path ,[object Object]

Ajani’s List: The Eastern Path ,[object Object],Eagle Rock

Ajani’s List: The Eastern Path ,[object Object],[object Object],Eagle Rock

Ajani’s List: The Eastern Path ,[object Object],[object Object],Eagle Rock 9km

Ajani’s List: The Eastern Path ,[object Object],[object Object],[object Object],Eagle Rock 9km

Ajani’s List: The Eastern Path ,[object Object],[object Object],[object Object],70 ° 9km Eagle Rock

[object Object],[object Object],[object Object],[object Object],70 ° 9km Ajani’s List: The Eastern Path Eagle Rock

70 ° 9km Ajani’s List: Other Knowns Eagle Rock ,[object Object]

70 ° 9km 20km Ajani’s List: Other Knowns Eagle Rock ,[object Object]

Step 1: Label the diagram 70 ° 9km 20km Eagle Rock Solving the Triangle

Step 1: Label the diagram 70 ° 9km 20km Eagle Rock Solving the Triangle a b c A B C

Step 1: Label the diagram Step 2: Solve for Angle A 70 ° 9km 20km Eagle Rock Solving the Triangle a b c A B C

Something to Think About; The total degree measurement of a straight line is 180°

Something to Think About; 180° The total degree measurement of a straight line is 180°

Step 1: Label the diagram Step 2: Solve for Angle A Since a straight line is equal to 180 ° 70 ° 9km 20km Eagle Rock Solving the Triangle a b c A B C

Step 1: Label the diagram Step 2: Solve for Angle A Since a straight line is equal to 180 ° Angle A = 180 ° - 70 ° Angle A = 110 ° 70 ° 9km 20km Eagle Rock Solving the Triangle a b c A B C

Step 1: Label the diagram Step 2: Solve for Angle A Since a straight line is equal to 180 ° Angle A = 180 ° - 70 ° Angle A = 110 ° 70 ° 9km 20km Eagle Rock Solving the Triangle 110 ° a b c A B C

There are two equations that may be used to solve for non-right angle triangles: Something to Think About;

There are two equations that may be used to solve for non-right angle triangles: Something to Think About; The Sine Law Equation: ,[object Object],[object Object],[object Object]

There are two equations that may be used to solve for non-right angle triangles: Something to Think About; ,[object Object],[object Object],The Cosine Law Equation:

Step 1: Label the diagram Step 2: Solve for Angle A Step 3: Solve for Angle C 70 ° 9km 20km Eagle Rock Solving the Triangle 110 ° a b c A B C

Step 3: Solve for Angle C Side c is known therefore Angle C can be solved for with the Sine Equation 70 ° 9km 20km Eagle Rock Solving the Triangle 110 ° a b c A B C

Step 3: Solve for Angle C Side c is known therefore Angle C can be solved for with the Sine Equation 70 ° 9km 20km Eagle Rock Solving the Triangle 110 ° To solve for angle C cross multiply: a b c A B C

Step 3: Solve for Angle C Side c is known therefore Angle C can be solved for with the Sine Equation 70 ° 9km 20km Eagle Rock Solving the Triangle 110 ° To solve for angle C cross multiply: And divide by 20 to isolate Sin C: a b c A B C

Step 3: Solve for Angle C Side c is known therefore Angle C can be solved for with the Sine Equation 70 ° 9km 20km Eagle Rock Solving the Triangle 110 ° To solve for angle C cross multiply: And divide by 20 to isolate Sin C: Solve and take the inverse Sin to calculate the angle. a b c A B C

Step 3: Solve for Angle C Side c is known therefore Angle C can be solved for with the Sine Equation 70 ° 9km 20km Eagle Rock Solving the Triangle 110 ° To solve for angle C cross multiply: And divide by 20 to isolate Sin C: Solve and take the inverse Sin to calculate the angle. 25.02° a b c A B C

Step 1: Label the diagram Step 2: Solve for Angle A Step 3: Solve for Angle C Step 4: Solve for Angle B 70 ° 9km 20km Eagle Rock Solving the Triangle 110 ° 25.02° a b c A B C

The sum of all three angles in a triangle is equal to 180° Something to Think About; A B C Therefore: A+B+C=180°

Step 4: Solve for Angle B Angle B can be solved for by subtracting Angles A and C from 180 ° 70 ° 9km 20km Eagle Rock Solving the Triangle 110 ° 25.02° a b c A B C

Step 4: Solve for Angle B Angle B can be solved for by subtracting Angles A and C from 180 ° Angle B = 180° - (110° + 25.02°) = 44.98° 70 ° 9km 20km Eagle Rock Solving the Triangle 110 ° 25.02° 44.98° a b c A B C

Step 1: Label the diagram Step 2: Solve for Angle A Step 3: Solve for Angle C Step 4: Solve for Angle B Step 5: Solve for Side b 70 ° 9km 20km Eagle Rock Solving the Triangle 110 ° 25.02° 44.98° a b c A B C

Step 5: Solve for Side b Now that angle B is known side b can be calculated using the Sine Equation. 70 ° 9km 20km Eagle Rock Solving the Triangle 110 ° 25.02° 44.98° a b c A B C

It is always best to use information originally given in a question to solve for other unknowns Something to Think About; ,[object Object],[object Object]

Step 5: Solve for Side b Now that angle B is known side b can be calculated using the Sine Equation. 70 ° 9km 20km Eagle Rock Solving the Triangle 110 ° 25.02° Cross multiple and divide to isolate b: 44.98° a b c A B C

Step 5: Solve for Side b Now that angle B is known side b can be calculated using the Sine Equation. 70 ° 9km 20km Eagle Rock Solving the Triangle 110 ° 25.02° Cross multiple and divide to isolate b: 15.04 km 44.98° a b c A B C

Ajani’s List: Other Knowns ,[object Object],70 ° 9km 20km Eagle Rock 110 ° 25.02° 15.04 km 44.98° a b c A B C

Ajani’s List: Other Knowns ,[object Object],70 ° 9km 20km Eagle Rock 110 ° 25.02° 15.04 km 72 ° 44.98° a b c A B C

70 ° 9km 20km Eagle Rock 110 ° 25.02° 15.04 km 72 ° 44.98° Step 1: Label the diagram Step 2: Solve for Angle A Step 3: Solve for Angle C Step 4: Solve for Angle B Step 5: Solve for Side b Step 6: Re-label new triangle Solving the Triangle a b c A B C

70 ° 9km 20km Eagle Rock 110 ° 25.02° 15.04 km 72 ° 44.98° ,[object Object],[object Object],[object Object],[object Object],Solving the Triangle a b c A B C

70 ° 9km 20km Eagle Rock 110 ° 25.02° 15.04 km 72 ° 44.98° ,[object Object],[object Object],[object Object],[object Object],[object Object],Solving the Triangle a b c A B C

70 ° 9km 20km Eagle Rock a b c A B C 110 ° 25.02° 15.04 km 72 ° 44.98° ,[object Object],[object Object],[object Object],[object Object],[object Object],Solving the Triangle

70 ° 9km 20km Eagle Rock a b c A B C 110 ° 25.02° 15.04 km 72 ° 44.98° Step 1: Label the diagram Step 2: Solve for Angle A Step 3: Solve for Angle C Step 4: Solve for Angle B Step 5: Solve for Side b Step 6: Re-label new triangle Step 7: Solve for Angle B Solving the Triangle

Step 7: Solve for Angle B Solving the Triangle 70 ° 9km 20km Eagle Rock a b c A B C 110 ° 25.02° 15.04 km 72 ° 44.98°

70 ° 9km 20km Eagle Rock a b c A B C 110 ° 25.02° 15.04 km 72 ° 44.98° Step 7: Solve for Angle B Both the 72° and 44.98 ° angles begin on the same axis then: Solving the Triangle

70 ° 9km 20km Eagle Rock a b c A B C 110 ° 25.02° 15.04 km 72 ° 44.98° Step 7: Solve for Angle B Both the 72° and 44.98 ° angles begin on the same axis then: Angle B= 72° - 44.98° Solving the Triangle

70 ° 9km 20km Eagle Rock a b c A B C 110 ° 25.02° 15.04 km 72 ° 44.98° Step 7: Solve for Angle B Both the 72° and 44.98 ° angles begin on the same axis then: Angle B= 72° - 44.98° Angle B= 27.02° Solving the Triangle 27.02 °

When a transversal line passes through two parallel lines corresponding angles are formed. Something to Think About;

When a transversal line passes through two parallel lines corresponding angles are formed. This means that the angles on alternate sides of the transversal are the same. Something to Think About;

Step 1: Label the diagram Step 2: Solve for Angle A Step 3: Solve for Angle C Step 4: Solve for Angle B Step 5: Solve for Side b Step 6: Re-label new triangle Step 7: Solve for Angle B Step 8: Solve for Angle C Solving the Triangle 70 ° 9km 20km Eagle Rock a b c A B C 72 ° 44.98° 27.02 ° 15.04 km

Step 8: Solve for Angle C Side a is a transversal line Solving the Triangle 70 ° 9km 20km Eagle Rock a b c A B C 72 ° 44.98° 27.02 ° 15.04 km

Step 8: Solve for Angle C Side a is a transversal line This means that Angle C is congruent to 44.98° Solving the Triangle 70 ° 9km 20km Eagle Rock a b c A B C 72 ° 44.98° 27.02 ° 15.04 km

Step 8: Solve for Angle C Side a is a transversal line This means that Angle C is congruent to 44.98° this is because it is on the alternate side of the transversal line to the original 44.98°. Solving the Triangle 70 ° 9km 20km Eagle Rock a b c A B C 72 ° 44.98° 27.02 ° 15.04 km 44.98°

Step 1: Label the diagram Step 2: Solve for Angle A Step 3: Solve for Angle C Step 4: Solve for Angle B Step 5: Solve for Side b Step 6: Re-label new triangle Step 7: Solve for Angle B Step 8: Solve for Angle C Step 9: Solve for Angle A Solving the Triangle 70 ° 9km 20km Eagle Rock a b c A B C 72 ° 44.98° 27.02 ° 15.04 km

Step 9: Solve for Angle A Since two angles are known: Solving the Triangle 70 ° 9km 20km Eagle Rock a b c A B C 72 ° 27.02 ° 15.04 km 44.98°

Step 9: Solve for Angle A Since two angles are known: Angle A = 180 ° - (44.98° + 27.02°) Solving the Triangle 70 ° 9km 20km Eagle Rock a b c A B C 72 ° 27.02 ° 15.04 km 44.98°

Step 9: Solve for Angle A Since two angles are known: Angle A = 180 ° - (44.98° + 27.02°) Angle A = 108° Solving the Triangle 70 ° 9km 20km Eagle Rock a b c A B C 72 ° 27.02 ° 15.04 km 44.98° 108°

Step 1: Label the diagram Step 2: Solve for Angle A Step 3: Solve for Angle C Step 4: Solve for Angle B Step 5: Solve for Side b Step 6: Re-label new triangle Step 7: Solve for Angle B Step 8: Solve for Angle C Step 9: Solve for Angle A Step 10: Solve for Side b Solving the Triangle 70 ° 9km 20km Eagle Rock a b c A B C 72 ° 27.02 ° 15.04 km 108° 44.98°

Step 10: Solve for Side b The Sine equation can be used to solve side b: Solving the Triangle 70 ° 9km 20km Eagle Rock a b c A B C 72 ° 27.02 ° 15.04 km 108° 44.98°

Step 10: Solve for Side b The Sine equation can be used to solve side b: Solving the Triangle 70 ° 9km 20km Eagle Rock a b c A B C 72 ° 27.02 ° 15.04 km 108° Cross multiply and divide to isolate b 44.98°

Step 10: Solve for Side b The Sine equation can be used to solve side b: Solving the Triangle 70 ° 9km 20km Eagle Rock a b c A B C 72 ° 27.02 ° 15.04 km 108° Cross multiply and divide to isolate b 44.98° 9.55km

Step 1: Label the diagram Step 2: Solve for Angle A Step 3: Solve for Angle C Step 4: Solve for Angle B Step 5: Solve for Side b Step 6: Re-label new triangle Step 7: Solve for Angle B Step 8: Solve for Angle C Step 9: Solve for Angle A Step 10: Solve for Side b Step 11: Solve the Eastern Route Solving the Triangle 70 ° 9km 20km a c A B C 72 ° 27.02 ° 15.04 km 108° 44.98° Eagle Rock b 9.55km

Step 11: Solve the Eastern Route To Solve for the entire distance of the eastern route: Solving the Triangle 70 ° 9km 20km a c A B C 72 ° 27.02 ° 15.04 km 108° 44.98° Eagle Rock b 9.55km

Step 11: Solve the Eastern Route To Solve for the entire distance of the eastern route: Eastern Route= 9km+15.04km+9.55km Solving the Triangle 70 ° 9km 20km a c A B C 72 ° 27.02 ° 15.04 km 108° 44.98° Eagle Rock b 9.55km

Step 11: Solve the Eastern Route To Solve for the entire distance of the eastern route: Eastern Route= 9km+15.04km+9.55km Eastern Route= 33.59km Solving the Triangle 70 ° 9km 20km a c A B C 72 ° 27.02 ° 15.04 km 108° 44.98° Eagle Rock b 9.55km

[object Object],70 ° 20km a c A B C 72 ° 27.02 ° 108° 44.98° Eagle Rock b Ajani’s List: The Western Path

[object Object],70 ° 20km a c A B C 72 ° 27.02 ° 108° 44.98° Eagle Rock b Ajani’s List: The Western Path 50°

[object Object],[object Object],20km a c A B C 72 ° 27.02 ° 108° 44.98° Eagle Rock b Ajani’s List: The Western Path 50°

Step 1: Solve Side c 20km a c A B C 72 ° 27.02 ° 108° 44.98° Eagle Rock b 50° Solving the Triangle

Something to Think About; a c A B C b ,[object Object],[object Object]

Something to Think About; a c A B C b ,[object Object],[object Object],[object Object]

Step 1: Solve Side c 20km a c A B C 72 ° 27.02 ° 108° 44.98° Eagle Rock b 50° Solving the Triangle Side c can be solved using the Sine Formula and values from the previous triangle:

Step 1: Solve Side c 20km a c A B C 72 ° 27.02 ° 108° 44.98° Eagle Rock b 50° Solving the Triangle Side c can be solved using the Sine Formula and values from the previous triangle: Cross multiply and divide to isolate c

Step 1: Solve Side c 20km a c A B C 72 ° 27.02 ° 108° 44.98° Eagle Rock b 50° Solving the Triangle Side c can be solved using the Sine Formula and values from the previous triangle: Cross multiply and divide to isolate c 14.86km c= 14.864729… = 14.86km

Step 1: Solve Side c Step2: Solve Angle A 20km c A B 72 ° 27.02 ° 108° 44.98° Eagle Rock 50° Solving the Triangle 14.86km a C b

Step2: Solve Angle A Angle A is on the same horizontal line as the Angle of 108° 20km c A B 72 ° 27.02 ° 108° 44.98° Eagle Rock 50° Solving the Triangle 14.86km a C b

Step2: Solve Angle A Angle A is on the same horizontal line as the Angle of 108° Therefore Angle A = 180 °-108 ° 20km c A B 72 ° 27.02 ° 108° 44.98° Eagle Rock 50° Solving the Triangle 14.86km a C b

Step2: Solve Angle A Angle A is on the same horizontal line as the Angle of 108° Therefore Angle A = 180 °-108 ° Angle A = 72° 20km a c A B C 72 ° 27.02 ° 108° 44.98° Eagle Rock b 50° Solving the Triangle 14.86km 72°

Step 1: Solve Side c Step 2: Solve Angle A Step 3: Solve Angle B 20km c A B 72 ° 27.02 ° 108° 44.98° Eagle Rock 50° Solving the Triangle 14.86km a C b 72°

Step 3: Solve Angle B Since the angles of 50° and 72° are both located on the x axis subtracting these from 180° will give the value of Angle B. 20km c A B 72 ° 108° 44.98° Eagle Rock 50° Solving the Triangle 14.86km a C b 72°

Step 3: Solve Angle B Since the angles of 50° and 72° are both located on the x axis subtracting these from 180° will give the value of Angle B. Angle B= 180° - (50°+72°) 20km c A B 72 ° 108° 44.98° Eagle Rock 50° Solving the Triangle 14.86km a C b 72°

Step 3: Solve Angle B Since the angles of 50° and 72° are both located on the x axis subtracting these from 180° will give the value of Angle B. Angle B= 180° - (50°+72°) Angle B= 58° 20km a c A B C 72 ° 108° 44.98° Eagle Rock b 50° Solving the Triangle 14.86km 58° 72°

Step 1: Solve Side c Step 2: Solve Angle A Step 3: Solve Angle B Step 4: Solve Angle C 20km a c A B C 72 ° 108° 44.98° Eagle Rock b Solving the Triangle 14.86km 58° 72°

Step 4: Solve Angle C Since we know a triangle is equal to 180° Angle C= 180° - (58°+72°) 20km a c A B C 72 ° 108° 44.98° Eagle Rock b Solving the Triangle 14.86km 72° 58°

Step 4: Solve Angle C Since we know a triangle is equal to 180° Angle C= 180° - (58°+72°) Angle C= 50° 20km a c A B C 72 ° 108° 44.98° Eagle Rock b Solving the Triangle 14.86km 72° 58° 50°

Step 4: Solve Angle C Since we know a triangle is equal to 180° Angle C= 180° - (58°+72°) Angle C= 50° 20km a c A B C 72 ° 108° 44.98° Eagle Rock b Solving the Triangle 14.86km 72° 58° 50° **Note to check if your angles are correct: Angle B plus 50° should equal 108 ° because it is on the alternate side of the transversal c. 50 ° + 58 ° = 108 ° 108°

Step 1: Solve Side c Step 2: Solve Angle A Step 3: Solve Angle B Step 4: Solve Angle C Step 5: Solve side a 20km a c A B C 72 ° 108° 44.98° Eagle Rock b Solving the Triangle 14.86km 58° 72°

Step 5: Solve side a 20km a c A B C 72 ° 108° 44.98° Eagle Rock b Solving the Triangle 14.86km 58° 72°

Step 5: Solve side a 20km a c A B C 72 ° 108° 44.98° Eagle Rock b Solving the Triangle 14.86km 58° 72° Cross multiply and divide to isolate a

Step 5: Solve side a 20km a c A B C 72 ° 108° 44.98° Eagle Rock b Solving the Triangle 14.86km 58° 72° Cross multiply and divide to isolate a a= 18.448929… = 18.45km 18.45km

Step 1: Solve Side c Step 2: Solve Angle A Step 3: Solve Angle B Step 4: Solve Angle C Step 5: Solve Side b 20km a c A B C 72 ° 108° 44.98° Eagle Rock b Solving the Triangle 14.86km 58° 72° 18.45km

Step 5: Solve Side b 20km a c A B C 72 ° 108° 44.98° Eagle Rock b Solving the Triangle 14.86km 58° 72° 18.45km

Step 5: Solve Side b 20km a c A B C 72 ° 108° 44.98° Eagle Rock b Solving the Triangle 14.86km 58° 72° Cross multiply and divide to isolate b 18.45km

Step 5: Solve Side b 20km a c A B C 72 ° 108° 44.98° Eagle Rock b Solving the Triangle 14.86km 58° 72° Cross multiply and divide to isolate b b= 16.450735… = 16.45km 16.45km 18.45km

20km a c A B C 72 ° 108° 44.98° Eagle Rock b Solving for the Western Route 14.86km 58° 72° 16.45km 18.45km To solve for the Western Route from the village to Eagle Rock:

20km a c A B C 72 ° 108° 44.98° Eagle Rock b Solving for the Western Route 14.86km 58° 72° 16.45km 18.45km To solve for the Western Route from the village to Eagle Rock: Add the two paths: Western Route= 18.45+16.45

20km a c A B C 72 ° 108° 44.98° Eagle Rock b Solving for the Western Route 14.86km 58° 72° 16.45km 18.45km To solve for the Western Route from the village to Eagle Rock: Add the two paths: Western Route= 18.45+16.45 Western Route= 34.90

20km a c A B C 72 ° 108° 44.98° Eagle Rock b Choosing the Shortest Route 14.86km 58° 72° 16.45km 18.45km 15.04 km 9.55km

20km a c A B C 72 ° 108° 44.98° Eagle Rock b Choosing the Shortest Route 14.86km 58° 72° 16.45km 18.45km 15.04 km 9.55km Route One = 9km + 15.04km + 9.55km = 33.59km

20km a c A B C 72 ° 108° 44.98° Eagle Rock b Choosing the Shortest Route 14.86km 58° 72° 16.45km 18.45km 15.04 km 9.55km Route One = 9km + 15.04km + 9.55km = 33.59km Route Two= 18.45km + 16.45km = 34.90km

20km a c A B C 72 ° 108° 44.98° Eagle Rock b Choosing the Shortest Route 14.86km 58° 72° 16.45km 18.45km 15.04 km 9.55km Route One = 9km + 15.04km + 9.55km = 33.59km Route Two= 18.45km + 16.45km = 34.90km According to Ajani’s calculations route one would be the shortest path to Eagle Rock.

Speed Ajani needs to know if he walks at a speed of 1.5km per hour and is planning to spend only 12 hours of each day of the two days he has walking to complete the journey how long will it take to travel the 33.59 km?

Speed Ajani needs to know if he walks at a speed of 1.5km per hour and is planning to spend only 12 hours of each day of the two days he has to complete the journey how long will it take to travel the 33.59 km? Step one: set up a ratio to use to calculate the speed Since Ajani is going to be walking for 12 hours each day for two days you need to know how far he will be able to walk in 24 hours. Cross multiply and divide to isolate the value of the km. Km= 36km Since Ajani only has to travel 33.59km he will be able to walk at a normal pace since he will be able to cover 36km in 24 hours.

Speed Ajani needs to know if he walks at a speed of 1.5km per hour and is planning to spend only 12 hours of each day of the two days he has to complete the journey how long will it take to travel the 33.59 km? Step one: set up a ratio to use to calculate the speed

Speed Ajani needs to know if he walks at a speed of 1.5km per hour and is planning to spend only 12 hours of each day of the two days he has to complete the journey how long will it take to travel the 33.59 km? Step one: set up a ratio to use to calculate the speed Since Ajani is going to be walking for 12 hours each day for two days you need to know how far he will be able to walk in 24 hours.

Speed Ajani needs to know if he walks at a speed of 1.5km per hour and is planning to spend only 12 hours of each day of the two days he has to complete the journey how long will it take to travel the 33.59 km? Step one: set up a ratio to use to calculate the speed Since Ajani is going to be walking for 12 hours each day for two days you need to know how far he will be able to walk in 24 hours. Cross multiply and divide to isolate the value of the km. Km= 36km

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