Every point in space is colored red, green, or blue. Prove or disprove that there exists a rectangular parallelipiped in space such that all of its vertices are the same color.

Q: 4. Give a geometric description of each family of planes.

A:

Q: Find the volume of the parallelepiped with one vertex at the origin and adjacent vertices at (1, 3,…

A: Consider the provided question, Let the vertices of the parallelepiped are A, B, C and D.

Q: Find T, N and k for the spacecurves:

A: We have to find Unit Tangent Vector T,  Unit Normal Vector N and Curvature K for given space curve…

Q: Give an example of three planes,exactly two of which are parallel.

A: Give an example of three planes,exactly two of which are parallel.

Q: Verify that the points are the vertices of a parallelogram, and find its area. A(1, 1, 3), B(8, -2,…

A: FIND THE AREA

Q: Verify that the points are the vertices of a parallelogram, and then find its area.(1, −2, 0), (4,…

A:

Q: Let ABCD be a general, not necessarily planar, quadrilateral in space. Show that the two segments…

A: Let ABCD be a quadrilateral.

Q: e of the parallelepiped with the given vertices A, B, C and

A:

Q: Prove that the diagonals of a parallelogram bisect each other.

A:

Q: In an artistic design by a contractor of the students’ centre in a university, four pillars were…

A: Given: In an artistic design, four pillars were erected to form two triangular figures with the…

Q: Verify that the points are the vertices of a parallelogram, and find its area. A(1, 1, 3), B(-8, 5,…

A:

Q: Prove that a quadrilateral with vertices J(2,-1), K(-1,-4), L(-4,-1), and M(-1,2) is a square.

A: We have to prove that the quadrilateral given is a Square

Q: Prove that if DABCD is a parallelogram, then DABCD is convex.

A:

Q: Let ABCD be a general, not necessarily planar, quadrilateral in space. Show that the two segments…

A: Given ABCD is a quadrilateral

Q: Prove that the diagonals of a rectangle are of equal length. [Hint: Take the vertices of the…

A: To prove that diagonals of a rectangle are if equal length.  Consider a rectangle which has one…

Q: Suppose that a box has its faces parallel to the coordinate planes and the points (4, 2,−2) and (−6,…

A:

Q: Verify that the points are the vertices of a parallelogram, and find its area. A(1, 1, 3), В(-4, -5,…

A:

Q: The directed line segment whose initial point is the origin is in ________ ________.

A: Our Aim is to tell what the directed line segment is being called whose initial point is the origin.

Q: Show that the points (−1, 0), (3, 0), (1, 2), and (1, −2) are vertices of a square.

A: Let us consider A(-1,0) , B(3,0), C(1,2) and D(1,-2) To verify ABCD as a square We need to verify if…

Q: Verify that the points are vertices of a parallelogram and find its area A(2,-3,1) B(6,5,-1)…

A:

Q: Find the areas of the parallelograms whose vertices are given A(1, 0), B(0, 1), C(-1, 0), D(0, -1)

A:

Q: Do the three planes x1 + 2x2 + x3 = 4, x2 – x3 = 1, and x1 + 3x2 = 0 have at least one common point…

A: The three planes x1+2x2+x3=4, x2-x3=1, and x1+3x2=0.

Q: Find the volume of a parallelepiped if four of its eight vertices are A(0, 0, 0), B(1, 2, 0), C(0,…

A: Draw the diagram roughly

Q: Prove that the Chebyshev center of an acute triangle in the Euclidean plane is the center of the…

A: Let I be an inversion with respect to c. Let denote A'=I(A). If the points A,A' and B are collinear.…

Q: prove that the midpoints of the sides of a quadrilateral of the mice are vertices of a…

A: We need to prove the midpoints of the sides of a quadrilateral of are vertices of a parallelogram.…

Q: Verify that the points are the vertices of a parallelogram, and find its area. А(1, 1, 3), в(7, -3,…

A:

Q: What is the locus of points in space equidistant from two parallel planes?

A: Locus: If the two lines are in Rn the points equidistant from them form an n−1 dimensional…

Q: Let ABCD be a general, not necessarily planar, quadrilateral in space. Show that the two segments…

A:

Q: Verify that the points are the vertices of a parallelogram, and find its area. А(1, 1, 3), B(—2, 3,…

A:

Q: Prove that there exists a set of two lines containing all the points of the geometry.

A:

Q: Let A, be the center of the square inscribed in acute triangle ABC with two vertice side BC (as…

A:

Q: Every point in the plane is colored using one of 2020 colors. Prove or disprove that there exists a…

A: Using the pigeon hole principle let's prove that can find a rectangle with vertices of same color.…

Q: Prove that the diagonals of any parallelogram bisect each other. (Hint: Label three of the vertices…

A: Given:- ABCD is a parallelogramTo prove:-AO=CO                 DO=BOProof:-In ∆AOD and ∆COB∡OAD=∡OCB…

Q: Find the perimeter of the triangle whose vertices are the following specified points in the plane.…

A:

Q: Verify that the points are the vertices of a parallelogram, and find its area. А(1, 1, 3), В(6, 2,…

A:

Q: Verify that the points are the vertices of a parallelogram, and find its area. A(1, 1, 3), B(10, -7,…

A:

Q: Prove that cyclic parallelogram is a rectangle.

A:

Q: All polygons are topologically equivalent. True False

A:

Q: Prove analytically that the midpoint of the hypotenuse of a right triangle is equidistant from the…

A:

Q: Verify that the points are the vertices of a parallelogram, and find its area. A(1, 1, 3), B(7, 2,…

A:

Q: 2) Determine the minimum number of degrees of rotation that would map each polygon onto itself. a)…

A:

Q: Prove that in the hyperbolic plane, given a positive integer n there exist n + 1 lines such that…

A: In the hyperbolic plane, given a positive integer n there exist n+1 lines such that earth sit of n…

Q: Every parallelogram is a rectangle. О rue O False

A:

Q: Verify that the points are the vertices of a parallelogram, and find its area. A(1, 1, 3), B(7, -3,…

A:

Q: Prove that if ABCD is a parallelogram, then its diagonals bisect each other.

A: Let us consider a paralellogram ABCD which have diagonals AC And BD and intersect at point E

Q: Show that A(-4, – 1), B(0, –2), C(6, 1), and D(2, 2) are vertices of a parallelogram.

A: Given  A(-4,-1), B(0,-2), C(6,1) and D(2,2) To show: ABCD are vertices of a parallelogram.

Q: Let ABCD be a convex quadrilateral with ZA = ZB = 120°. Prove that CD² = (AB + AD)² + (AB + BC)² –…

A:

Q: Prove that if two pairs of opposite edges of a tetrahedron are perpendicular, then the third pair is…

A: Consider the tetrahedron  ABCD and let two pairs of opposite sides are perpendicular. Consider the…