Geometry / Mr. Hansen 5/8/2003

Name: _________________________ Test #12

Test on Chapter 13

	Part I: Enhanced Always, Sometimes, Never (3 pts. each). Write the letter A, S, or N in each blank. Work is not required, but if you provide a worthy diagram or a short explanation, you can earn partial credit even if your answer is incorrect.
1. ___	The slope of a line equals a real number.
2. ___	The slope of a horizontal line is zero.
3. ___	If the coordinates of a point in an xyz coordinate system are (q, s, t), then the square of the point’s distance from the origin equals q^2 + s^2 + t^2.
4. ___	Two distinct planes that are not parallel intersect at a single point.
	Part II: Creative Work (10 pts.).
5.	Write a problem that involves a circle and the slope of a line of tangency. Be sure to state clearly what the “givens” of the problem are, and what it is that the student is supposed to find. Then solve your problem, showing your work. You can make your problem fairly easy, as long as the statement and solution are clear.

	Part III: Problems (10 pts. each). Although many of these can be solved by inspection, you must make at least a rough sketch in each case to earn full credit. Circle your answer. When a numeric answer is requested, give answer either in simple radical form or with at least 3 decimal places of accuracy. Include units if appropriate. Show your work legibly and completely. If you run out of time, describe how you would solve the problem if you had more time. Initial here if you understand these conditions: ___ (2 pts.)
6.	Find an equation (any form is acceptable) of the line having x-intercept of –2 and passing through (4, –5). Remember the instructions that you initialed above for this section.
7.	Compute the distance between (–1, 1, 1) and (3, 5, 4). Remember that a sketch is required for this and all other problems in this section. (Label your axes, and please make your z-axis be the one that comes “out of the paper” toward you.)
8.	A circle has center C(–2, 2) and passes through the point P(–1, –5).
(a)	Make a sketch (as you must for all problems in this section).
(b)	Write an equation of the circle.
(c)	Use your result from part (b) to show that P is on the circle.
(d)	Is the origin inside the circle? _______ How do you know? _____________________________

9.	Find the solution of the following system both graphically and algebraically. Your graphic solution can be very rough—showing the proper quadrant is sufficient. However, your algebraic solution should be exact and should be given in the form of a solution set. You may use any algebraic procedure you wish, as long as it is clear. 2x + y = 7 x – 2y = 10
10.	The equation x^2 + 2y = 3x – y^2 + 73¾ can be plotted as a circle. Complete the square(s), find the center, radius, and area, and make a rough sketch. Center = __________ , r = __________ , A = __________ .
11.	Consider the system x^2 + y^2 ³ 9 y < x + 2 Sketch these inequalities on the same set of axes, using a different style of shading for each. Use “double shading” to indicate the solution set. No algebra is required.

	Part IV. Multiple Choice (3 pts.)
12.	We discussed the difference between (I) true mathematical theorems, (II) scientific theories, and (III) nonscientific theories. Which of these are potentially provable? Which are potentially disprovable?
	(A) Provable: I only. Disprovable: II only.
	(B) Provable: I and II only. Disprovable: II and III only.
	(C) Provable: I only. Disprovable: II and III only.
	(D) Provable: I, II, and III. Disprovable: III only.
	(E) Provable: I and II only. Disprovable: III only.
	Part V. Proof (12 pts., plus 1 for fill-in question below)
13.	There are two versions of this problem. One is for “number crunchers” and the other is for “conceptual thinkers.” Please decide which version you would like to work, and do it in the space below. Note that there are two things you must prove.
	Given: DABC, M midpt. of side AC, N midpt. of side BC Prove: sMN || sAB, and MN = ½AB
	“Number crunching” version: Let the vertices of DABC be A(–3, –2), B(8, –1), and C(6, 6). Make a diagram.
	“Conceptual thinking” version: Place DABC on a coordinate grid. Treat points A, B, C, M, N as unknown, but you may assume (wlog) that A is at the origin if you wish. This version is actually easier than the “number crunching” version, but it is a bit conceptual.
(1 pt.)	The conclusion is immediate by the ___________________________ that we saw earlier in the year. However, you must use coordinate geometry today.