3. Consider the real Cartesian plane with the standard distance function,d = the square root of [(x2 - X1)²+ (y2 - y1)²].Determine a corresponding coordinate system for the line I given by y = 2x + 3 (that is, a function f:l --> R such that for any two points P and Q on l, f (x, y) = \f(Q) - f(P)|.) Show all of your calculations.4. If fis a coordinate system on a line l, determine whether or not each of the following functions is a coordinate system on l.a. g(P) = f(P) - 10b. h(P) = 10f(P)с. К(Р) %3D 10 - f(P)d. r(P) = f(P) + the square root of 105. If P and Q are points and the Ruler Postulate is assumed to be true, prove the following.а. РQ 2 0b. PQ = 0 if and only if P = Qc. PQ = QP6. Determine whether each statement is true or false.a. The Poincaré Incidence Plane is an example of a projective plane.b. An incidence plane that is neither affine nor projective is hyperbolic.c. Lines in a plane satisfying the Incidence Axiom and Ruler Postulate have an infinite number of points.d. If theRuler Postulate is true, thethe Triangle Inequality.SOe. Assuming only the Incidence Axiom and Ruler Postulate, if lines l and m are both parallel to line n, then l is parallel to m.

Only first individual question is to be answered. Kindly repost the remaining questions.

3. Consider the real Cartesian plane with the standard distance function, d = the square root of [(x2 - x1)? + (y2 - Yı)?). Determine a corresponding coordinate system for the line I given by y = 2x + 3 (that is, a function f:l --> R such that for any two points P and Q on l, f (x, y) = [f(Q) - f(P)|) Show all of your calculations. 4. If fis a coordinate system on a line l, determine whether or not each of the following functions is a coordinate system on l.

3. Consider the real Cartesian plane with the standard distance function, d = the square root of [(x2 - x1)? + (y2 - Yı)?). Determine a corresponding coordinate system for the line I given by y = 2x + 3 (that is, a function f:l --> R such that for any two points P and Q on l, f (x, y) = [f(Q) - f(P)|) Show all of your calculations. 4. If fis a coordinate system on a line l, determine whether or not each of the following functions is a coordinate system on l.

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

SectionP.CT: Test

Problem 1CT

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Concept explainers

Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.

Question

3. Consider the real Cartesian plane with the standard distance function,
d = the square root of [(x2 - X1)²+ (y2 - y1)²].
Determine a corresponding coordinate system for the line I given by y = 2x + 3 (that is, a function f:l --> R such that for any two points P and Q on l, f (x, y) = \f(Q) - f(P)|.) Show all of your calculations.
4. If fis a coordinate system on a line l, determine whether or not each of the following functions is a coordinate system on l.
a. g(P) = f(P) - 10
b. h(P) = 10f(P)
с. К(Р) %3D 10 - f(P)
d. r(P) = f(P) + the square root of 10
5. If P and Q are points and the Ruler Postulate is assumed to be true, prove the following.
а. РQ 2 0
b. PQ = 0 if and only if P = Q
c. PQ = QP
6. Determine whether each statement is true or false.
a. The Poincaré Incidence Plane is an example of a projective plane.
b. An incidence plane that is neither affine nor projective is hyperbolic.
c. Lines in a plane satisfying the Incidence Axiom and Ruler Postulate have an infinite number of points.
d. If the
Ruler Postulate is true, the
the Triangle Inequality.
SO
e. Assuming only the Incidence Axiom and Ruler Postulate, if lines l and m are both parallel to line n, then l is parallel to m.

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