Prove that the following two statements (axioms) are equivalent: (i) If e is a line and P is a point not on , then there exists exactly one line k through P and parallel to e. (ii) If k, l and t are lines such that k || and t intersects k (so tnk|= 1) then t intersects (so |tn= 1).

Q: Given the following points A, B, C and D, which of the vectors and are parallel? CD (a) A= (1, 2),…

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Q: Determine the accuracy of the statements given below. X = 1- 2t I: x+ 2y+ 2z = 3 plane and y = 2+ 3t…

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Q: In the accompanying diagram, trans- versal RS intersects parallel lines +> XY and WZ at E and H,…

A: Given: RS is transversal line that means it intersects two parallel lines (XY and WZ). Therefore…

Q: mal ni 080 23. 2291 D 3x bni For hey A %23 B bag ABC is a straight line.

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Q: The Theorem of Pappus states that if A,B,C are three distinct points on one line and if A',B',C' are…

A: Given : The Theorem of Pappus states that if A,B,C are three distinct points on one line ...

Q: x+2 y+1 z-3 and 7 = [4,-2,1]+t[-2c,2,c] Determine the value of c that makes the lines 5…

A: The direction ratio of is:  (4, 5, 3)   The direction ratio of is:  (-2c, 2, c)

Q: Lines a and b are parallel if and only if slope of a = slope of b (or both a and b are vertical).…

A: Given that a and b are parallel if and only if slope of a = slope of b  21]a] Line l is parallel to…

Q: Gordon and Tess are trying to determine whether AABC and AFGE can be proven congruent through rigid…

A: From the given graph we can see that, if the coordinates of vertices in triangle ABC is (x,y) then…

Q: The distance between the line P(t) = (-4,-3,-5) +t(1,0, 2) %3D and the line Q(t) = (-3, -4,0) +t (2,…

A: The vector equation of a line passing through a point with position vector a1→ and parallel to a…

Q: Which of the following is/are axiom/s in projective geometry? I. All the points of the plane belong…

A: We have to determine which of the following is axiom of projective geometry.

Q: 16. Consider the points P(1, 2), Q(-2,1) and R(1,0) in the plane. Which one of the following…

A: 3 points are in the plane Which Statment is true ?

Q: Which of the following is the correct definition for perpendicular lines? O Lines that intersect and…

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Q: If m, n, and t are distinct lines in the same plane such that mIt and t In, then m n. O True O False

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Q: 2) Determine whether the lines 1: x = 1+ 2t, y = 3t, z = 2 – t and l2 : x = -1+ s, y = 4 + s, z…

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Q: Consider the following lines. L is the line through a= Lo is the line through = (1,0, 1) and d (3,…

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Q: Let l1 : (x, y, z) = (131, 89, −83) + r (42, 28, −28) where r ∈ R and l2 : (x, y, z) = (29, 21,…

A: LetandShow that the lines  and  are identical.

Q: Let l1, l2, and lz be nonconcurrent lines. Prove that there exist three points, exactly one each…

A: Given l1, l2, l3, three non cocurrent lines. the following cases are possible:  Case1: All lines are…

Q: Q.2) The horizontal line h in space placed in position that the coordinates of its two ends are…

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Q: There are infinitely many secant lines from an exterior point to a circle. True or False? O True…

A: From a point outside the circle, infinite secants can be drawn to the circle

Q: Let A be the point with coordinates (-2, -1, -1) and let B be the point with coordinates (2, 0, 0).…

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Q: Let P(6, – 2, 2) and Q(0, 0, 0) be two points in R³. Find the distance between P and Q. - PQ

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Q: Prove: In Young’s geometry, for every line l, there exists exactly two lines m and n such that m//l…

A: Given:  In Young’s geometry, for every line l, there exists exactly two lines m and n such that m//l…

Q: 20. Consider lines €1 and l2. l1 : [x, y] = [1, 4] + [[-7, 2] l2 : [x, y] = [-2, 6] + t[k, 9] a) For…

A: We know that two lines are parallel if their slope are equal and perpendicular if the product of…

Q: Prove that if the points (x1, y1) and (x2, y2) lie on the same line as (x∗1 , y∗1 ) and (x∗2 , y∗2…

A: Prove that if the points (x1, y1) and (x2, y2) lie on the same line as (x∗1 , y∗1 ) and (x∗2 , y∗2…