1. Prove that given a line é and a point P not on e, there exists a linethrough P perpendicular to €.2. If in AABC we have ZB LC, then AB AC and AABC isisosceles.3. State and prove the Alternate Interior Angle Theorem.4. State and prove the Exterior Angle Theorem.5. Prove that in a triangle, the greater angle lies opposite the greaterside and the greater side lies opposite the greater angle.6. Prove that the angle sum of every triangle is 180° if we assumeHilbert's Euclidean parallel postulate.7. How many points and lines are there in the Fano's Geometry? Justifyyour answer.8. Draw a representation of the four-line geometry. Name all the pointsand lines.9. Draw a representation of the four-point geometry. Name all thepoints and lines.10. Draw a representation of the Young's Finite Geometry. Name all thelines and determine which lines are parallel to cach other.

Explanation of the answer is as follows

Answered: 1. Prove that given a line € and a…

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Often we are interested in a point between two given points A and B which is not exactly at the midpoint. For 5, 2] and B = [1,6| , how can we find the point N which lies on the line segment AB, and is one example if A quarter of the way from A to B? This is the same as asking for the midpoint between A and the midpoint M = [3,4] of the line segment AB, namely rož +vof = No We could also write ON = }0Å 1 A+ 2 + 3 of the way from A to B in this case? It is given by 4 What is the point P, which is OP = ! OA+ OB) + 2 OB 1+ OB [2,5]
If S. F.dr is independent of the path C joining any two points, show that there exists a function O such that F= V$.
20. Consider lines ₁ and ₂. ₁: [x, y] = [1, 4] + [-7, 2] l₂: [x, y] = [-2, 6] + [k, 9] a) For what value of k are the lines parallel? b) For what value of k are the lines perpendicular? mine if each yvector equation € 3.
1. Every morming Neo packs his backpack and walks a distance L through a straight path in the forest from his home (A) to the unicorn sanctuary (B). One day he discovers someone has built an electric fence in the exact middle of his daily path (the dashed, red line in the picture): A• B The fence has length 2b - it extends for a distance b on each side of the path - and is perpendicular to the path. After a few days, Neo notices that the electric fence is turned on only half the time, but he does not know if the fence is on or off on any given day until he walks up to it and throws his cat at the fence to test it. If the fence is off, he can just quickly climb over it. Otherwise, he has to walk around it. He devises a plan: he will walk straight from his home to some point P in the fence; then, he will walk around it or climb over it depending on whether the fence is on or off. Which point P should he choose in order to minimize the average length of his trip?
QUESTION 8 Let m and n be two distinct lines. Suppose that Rn Rm (P)=P for some point P. Prove that Rn (P)=Rm (P). Attach File
5. Given: Any point D on a line / Construct: The perpendicular to / at D
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1. Prove that given a line € and a point P not on e, there exists a line through P perpendicular to e.