Homework # 1

Geometry and a Mini-Orchard

 

 

 

 

 

 

Fig #1 Orchard of Radius 1

Orchard, Early Spring.jpg

Part I: Summarizing Geometry

Here is your chance to excel. Summarize all the concepts and terms writing them down and making  index cards.

Be the first learning the concepts on the reference pages!

Part II: Mini Orchard

1. In what direction should you look?

This assignment is very simple. Just imagine that you are located in the center of that mini-Orchard. Imagine that those 4 trees are growing, and growing, and growing!

In what direction will you run if you want to leave before the growing trees block your path? Will you have more that one option?

What will happen if you don't get out on time?

Make sure to represent those option with a line starting on the center.

 

 

 

 

Do you own a compass?
You really need to buy one.

 

 

See what happened if the trees keep growing?

Mark the location where you think that the trees will meet.

What can you share with your group about those points?

The trees keep growing and growing.

 

 

 

 

 

 

 

2. What is the minimum radius required for each tree trunk in order to make the center of the orchard into a true "orchard Hideout"?

Since you did your homework, you are now ready to apply the Pythagorean Theorem. C2 = a2 + b2

Do you see a Right Triangle in this situation? Do you know the length of two sides? Can you find the missing side?

Remember: The distance from the center of the orchard to the center of any tree is ONE UNIT.

 

 

 

 

 

 

 

 

1

 

1

 

                                                     

More questions:

What do you think the red line represents?

Where on that line segment the two radii will meet?

If the length of the red line is d, what part of d is a radius?

The answer to these questions are very, very important.
 
Remember:
The question is "what is the minimal radius."
 Do you see a radius on the red line?
Can you use Pythagoras to find the distance of the hypotenuse?
How is the hypotenuse and the radii of the trees related?

C2 = a2 + b2

Please leave your answer as a fractions (we do not like long decimals) or as a radicals. Do you know what radicals are?

Please, let me know your final answer.

Your answer should look like:

The minimal radius required for each tree trunk in order to make the center of the orchard a true "orchard hideout is ________________________

Here is how I used The Pythagorean Theorem to find my answer.

_______________________Show your diagram here_____________________
 
Label your diagram

_________________________Show your work here_______________________

 

Make sure to organize your papers and have them ready  for school tomorrow.

Share your work with your group and ask as many questions as possible.

Other solutions
Using trigonometry: What is the measurement of the acute angles form on the right triangle?
If you know an angle and one side of a right triangle you can use one of the following trigonometric ratios (see reference pages):
Sine, Cosine, or tangent.
Which one relate the side = 1 and the distance that you are looking for?
Try it.
You should get the same answer!
 
Using area:  What is the area of the square form by connecting the centers of the four trees? Look at your diagram. It should be easy to find the area.
So, Area = d2
If you know the area of a square you can find the length of it side.
Try it.
You should get the same answer!
 
And finally by measuring: You may get a very good estimation by drawing your diagram to scale and measuring the length of d, but you should insist in getting the exact answer instead.
Try it.
You should get about the same answer!
 
 

If you do what you did today I guarantee you a A+ in this course.

Keep it up!

Mr. Lora

 

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