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Modern Geo 2
| Question | Answer |
|---|---|
| What's a Pencil? | A perspective image of a collection of parallel lines. That is, a collection of lines passing through a common point, or it could be a collection of parallel lines. |
| Define three collinear points. | Three points that share a line. |
| Define three coincident points. | Three lines that intersect at the same point. -Also known as concurrent points. |
| Define quadrilateral. | Four lines, six vertices. |
| Define three quadrangle. | Four vertices and connect them with six lines. |
| Let L and L' be 2 distinct lines a plane(w). Assume the point O lines in the plane w but not on either of these two lines. That is, P' is the intersection of lines L' and OP. a. Can we have no image for P'? b. Could P' be a line? c. Must P' be a point? | a.) No, unless something blocked the projection. b.) No, projections can only move "down" (i.e. line to a point.) c.) Yes, it can't be manipulated into a line or plane. |
| Let L and L' be 2 distinct lines a plane(w). Assume the point O lines in the plane w but not on either of these two lines. That is, P' is the intersection of lines L' and OP. d.) Is there a situation where a point on L' has no preimage? | d,) P with line L will always project onto P' through O, so no. |
| Let L and L' be 2 distinct lines a plane(w). Assume the point O lines in the plane w but not on either of these two lines. That is, P' is the intersection of lines L' and OP. e. What ensures that our projection will never project a point to a line/plane? | e.) Point P' intersects L and L', and two lines intersect at a point. |
| Let L and L' be 2 distinct lines a plane(w). Assume the point O lines in the plane w but not on either of these two lines. Suppose points A, B, and C are on line l in that order. Then must B' lie between A' and C'? | No. If A is placed behind O, with B and C being in front of O, then they will not always be in the order ABC. |
| Define in the most minimal way possible: a.) Equilateral triangle. b.) Isosceles triangle. c.) Square. | a.) Triangle with equal side lengths. b.) Triangle with two equal side lengths. c.) A quadrilateral with four congruent sides and four right angles. |
| Define in the most minimal way possible: d.) Right angle. e.) Circle. | d.) An angle formed from perpendicular lines. e.) The collection of points that are exactly radius (r) units away from the center. |
| True or False: There exists a pair of distinct lines k and L both in E^2 such that kL contains a single point. | True. |
| True or False: It is possible for a pair of distinct lines in E^2 to have two or more points of intersection. | False. |
| True or False: Every pair of distinct lines in E^2 intersects in one and only one point. | True. |
| True or False: If P is an ordinary point and Q is an ideal point, the PQ exists. ("PQ" is a line containing points P and Q). | True. |
| True or False: Every pair of distinct points P and Q in E^2 determines exactly one line (Here, "determines" means there is _________ and only _______ line that contains both _____ and _______) | True. There is one and only one line that contains both P and Q. |
| True or False: Every pair of distinct lines in E^3 determines exactly one line. | True. |
| True or False: Any three points in E^3 determine exactly one plane. | True. |
| True or False: Any two distinct intersecting ordinary lines determine exactly one plane. | True. |
| True or False: An ordinary line together with an ideal line that share a common point determine exactly one plane, | True. |
| True or False: Any two distinct intersecting ideal lines determine exactly one plane. | True. Forms an ideal plane. |
| True or False: Any two distinct ideal lines in E^3 intersect in one and only one point. | True. |
| True or False: Any two distinct lines in E^3 intersect in one and only one point. | False. Could be skew. |
| True or False: If a pair of distinct lines in E^3 is coplanar, then they intersect in a point. | True. |
| True or False: Two distinct lines in E^3 intersect in a point only if the lines are coplanar. | True. |
| True or False: If the line L on E^3 is not a subset of the plane a in E^3, the L intersecting a is exactly one point, | True. |
| True or False: Any two distinct planes in E^3 intersect in exactly one line. | True. |
| True or False: Any three planes in E^3 intersects in exactly one point. | False. Could intersect in a line. |
| Is it possible to have a triangle in E^2 with no ordinary points? If not, explain. | |
| Is it possible to have a triangle in E^2 with exactly one ordinary point? If not, explain. | |
| Is it possible to have a triangle in E^2 with exactly two ordinary points? If not, explain. | |
| Is it possible to have a triangle in E^2 with three ordinary points? If not, explain. | |
| If the three distinct vanishing lines of three planes a, b, and c intersect in a common point, what can you say about the lines formed by intersections of those planes? | |
| If three distinct vanishing lines of three planes a, b, and c don't pass through a common point, what can you say about the lines formed by intersections of those planes? | |
| Two distinct points in E^3 determine a unique _______. | Line. |
| Two distinct lines in E^3 line in the same_____ if and only if the lines ______ in exactly one ______. | a.) Plane b,) Intersect c.) Point |
| Two distinct planes in E^3 determine a unique ______. | Line. |
| A plane and a line not on the plane determine a unique ______. | Point. |
| A line and a point not on the line determine a unique _____. | Plane. |
Created by:
OliviaRoark
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