If x = 3 + ( 1 / (3 + (1 / x) ) and y = 3 + ( 1 / (3 + (1 / (3 + (1 / y) )) ), find abs(x - y).

 In Triangle ABC, a point D is placed on segment AC such that AB=AD (see figure 1-2 below).  If the measure of angle ABC minus the measure of angle ACB is 30 degrees, find the measure of angle CBD.

Sec 10.4 Enrichment

Use the given information and the diagrams provided to complete Exercises 1–4.

1.  Given: X, Y, and Z are the midpoints of the sides of DABC. Find the ratio of the area of DABC to the area of DXYZ.

2. 
Given: W, X, Y, and Z are the midpoints of the sides of square ABCD.

Find the ratio of the area of ABCD to the area of WXYZ.

3.  Given: R, S, T, U, V, and W are midpoints of the sides of regular
hexagon ABCDEF.

Find the ratio of the area of ABCDEF to the area of RSTUVW.

4.  Predict how the ratio of areas will change as the number of sides of the polygon increases.

Extra Credit with the Triangle Sum Theorem

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′