Math 445 Assignment 5

Math 445 Week 5

Assignments this week will be daily, not weekly, due to the upcoming midterm.

Review and exploration of affine coordinates on a line

Distances and ratios in a segment.

Draw a segment AB. Suppose |AB| = 5. If P is a point on segment AB with |AP| = 3, what is |BP|? What is |AP| + |BP|? What is AP/AB? What is BP/AB? What is the sum of these two ratios?

With the same A and B, suppose Q is a point on line AB but not on segment AB, with |AQ| = 3. What is |BQ|? Using signed ratios, what is AQ/BQ? What is BQ/AB? What is the sum of these two ratios?

On a different segment CD, suppose U is a point on segment CD with CU/CD = 1/6. What is UD/CD?

Locating a point on a line

Approach #1. Specify distances.

Suppose I want to tell you the location of a point R on a line CD. Draw a line CD on your paper.

If I tell you |CR| = 4, can you tell me where R is located?
If I also tell you |CD| = 10, can you tell me where R is located?
If in addition, I tell you |DR| > |CD|, can you tell me where R is located?
How many pieces of information did it take to draw the location of R?
If instead of 4, you were told |DR| = 15, could you plot R?

Approach #2. Specify ratio of distances.

Now, draw a line EF. Suppose I want to tell you the location of a point S on a line EF. Draw a line EF on your paper.

If I tell you |ES|/|EF| = 2/5, can you tell where S is located?
If I also tell you that |FS|/|EF| > 1, can you tell where S is located?
How many pieces of information did it take to draw the location of S?
If instead of 2, you were told FS|/|EF| = 8/5, could you plot S?

Approach #3. Specify signed ratio of distances.

Now, draw a line GH. Suppose I want to tell you the location of a point T on a line GH. Draw a line GH on your paper.

If I tell you GT/GH = -2/5, can you tell where T is located?
How many pieces of information did it take to draw the location of T?
In this case, if GT/GH = -2/5, what is TH/GH?

Assignment 5A. Due Wednesday (1/31)

[ See email list for details about turning in this assigment.]

Problem 5.1

In the real line, if point A = a, and point B = b, use algebra to find the point P (which = number p) on segment AB with AP/AB = t. The answer should be an expression made up of some real numbers and a, b, and t. Write your answer two ways. In one answer collect the "a" and "b" terms separately; in the other, collect the terms with t separately.

Problem 5.2

In the real coordinate 2-plane, if point A = (a1,a2), and point B = (b1,b2), draw an example of A and B and also the 4 coordinate projections on the axes (These are: A1= (a1,0), A2 – (0,a2), B1 = (b1,0), B2 = (0,b2).). Let P be a point on line AB. Draw an example of P = (p1,p2) and also the points P1 = (p1,0) and P2 = (0,p2).

Using what you know about parallels, if AP/AB = t (signed ratio), tell what is A1P1/A1B1 and A2P2/A2B2 and tell why.
Using your work in 5.1, write a formula for the coordinates of P1 and P2 in terms of t and the coordinates of A1 and B1 and A2 and B2.
Put this together to find a formula for P in terms of t, A and B. Write this formula with the coordinates explicitly, but also write it again as a vector formula.

Problem 5.3. Answer the Distances and Ratios in a Segment on the first page.

Problem 5.4. Answer the Locating a point on a Line on the page above.

Problem 5.5. Answer the Locating a point on a Plane questions posed in class and put on the class web site.

Assignment 5B. Due Monday (2/5)

Problem 5.6. Answer the Questions about Cubes posed in class that will be put on the web site.