THE CYLON!  WRITE THE SOLUTION AS A PROOF!

Forms of Logical Argument

The study of logic as a discipline has existed for thousands of years. Greek mathematicians and philosophers as far back as 600 BCE attempted to define how to argue logically. Two types of argument forms come from ancient times and have Latin names—modus ponens and modus tollens.

For Exercises 1–6, identify whether the argument is of the form modus ponens, modus tollens, or neither.

1.  If it rains in Spain, then it rains on the plain.

It is raining in Spain. Therefore, it is raining on the plain.

2.  If the scissor-tail swallows have returned, then it is not freezing outside.

It is freezing outside. Therefore, the scissor-tail swallows have not returned.

3.  You cannot get a speeding ticket if you do not speed.

You do not speed. Therefore, you cannot get a speeding ticket.

4.  If the animal is a dog, then the animal barks.

The animal barks, therefore it is a dog.

5.  If the train is on time, I will get to work on time.

If I get to work on time, I get to take a full hour for lunch.

I didn’t get to take a full hour for lunch. Therefore, the train wasn’t on time.

6.  If I think, then I am.

I think, therefore I am.

7.  Open-Ended Write an argument that uses modus ponens.

8.  Open-Ended Write an argument that uses modus tollens.

Three-Dimensional Rotations

Rotations in three dimensions are generally covered in calculus and analytical
geometry. However, a little deeper insight into rotations in two dimensions can be
gained by taking a brief glimpse at rotations in three dimensions.

Three-Dimensional Coordinate System

 

The three-dimensional coordinate system has three axes: x, y,
and z, as shown at the right. There are three intersecting planes,
each of which is referred to by the two axes that it contains:
the xy plane, the xz plane, and the yz plane. Each point in the
system has coordinates in the form (x, y, z).

In two dimensions, rotations are movements around a fixed
point in a plane. A triangle rotated about the origin in two
dimensions could have been “moved” just as easily by leaving
the triangle as is and rotating the x- and y-axes. In fact, in three
dimensions this is exactly how rotations are accomplished. The axes
are rotated, and the geometric figure is mapped to a new location.

For example, the rectangular solid below has vertices at A(–2, 0, 3), B(–2, 2, 3),
C(–2, 2, 0), D(–2, 0, 0), E(–6, 0, 0), F(–6, 0, 3), G(–6, 2, 3), and H(–6, 2, 0).
The second figure shows the solid mapped to a new position by a clockwise
90° rotation of the x- and y-axes.

 

 

What are the coordinates of the new vertices?

1.  A′	2.  B′	3.  C′	4.  D′
5.  E′	6.  F′	7.  G′	8.  H′

What will the coordinates be if the original rectangular solid is mapped to a new
position by a clockwise 180° rotation of the x- and y-axes?

9.  A′	10.   B′	11.   C′	12.   D′
13.   E′	14.   F′	15.   G′	16.   H′

Another from Nathan....

Given Triangle ABC where AC=BC, Angle ACB = 96 degrees, D is a point somewhere inside ABC such that Angle DAB = 18 degrees and Angle DBA = 30 degrees.  What is the measure (in degrees) of Angle ACD?

Thanks Nathan...

November Challenge

This one also comes from Nathan on the Geometry Math Team...


Prove that the LARGEST ANGLE of ANY TRIANGLE is ALWAYS GREATER THAN OR EQUAL TO 60 degrees.


I would recommend solving it by indirect proof.  Start by saying "What if the largest angle is less than 60? ...

	Chapter 4 Extra Credit Challange!

An art student wants to make a painting with a simple geometric pattern. She starts with a square. She divides this square into two congruent triangles. Then she divides each of these triangles into two smaller congruent triangles. She repeats the process seven more times. What does her pattern look like in the end?

1.  Show that the two triangles are congruent using the Hypotenuse-Leg Theorem.

2.  Use your knowledge of the Hypotenuse-Leg Theorem to divide each triangle in the figure above into two smaller congruent triangles. Repeat the process six more times.

3.  How do you know that the triangles at each step are congruent?

4.  How many triangles of the smallest size are shown?

5.  How many triangles are shown if they each contain 64 of the smallest-sized unit?

6.  How many triangles are shown if they each contain nine of the smallest-sized unit?

This one from Nathan from the Geometry Math Team.  See what you can do with it.