Point A is on the inside of a maze that is a simple closed curve. A segment from point A to the outside of the maze must cross the border of the maze an ___EVEN____ number of times for point A to be inside the maze.
If a convex polyhedron has 6 vertices and 8 faces, then it has __12____ edges.
The number of points in a taxicab circle with radius 5 is ___20___.
Which of the following is the only degree measure that could not be the sum of the angle measures of a spherical triangle?
220°
300°
182°
544° <-----
All correct except the first.
A point from the inside of a simple figure will cross an odd number of times to connect to any point outside.
Thank you very much MathMate :)
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To determine the number of times a segment from point A to the outside of the maze must cross the border of the maze, we can use the Jordan curve theorem. This theorem states that a simple closed curve (like the boundary of a maze) divides the plane into an inside and an outside region, and any segment from a point inside the curve to a point outside the curve must cross the curve an even number of times. Therefore, to find the number of times the segment must cross the border, we need an EVEN number.
To determine the number of edges in a convex polyhedron, we can use Euler's formula, which states that for any convex polyhedron, the number of vertices (V), edges (E), and faces (F) are related: V - E + F = 2. In this case, we are given that the polyhedron has 6 vertices and 8 faces. Plugging these values into Euler's formula, we can solve for the number of edges (E): 6 - E + 8 = 2. Solving this equation gives us E = 12, so the polyhedron has 12 edges.
To find the number of points in a taxicab circle with a given radius, we can use the formula for taxicab geometry. In taxicab geometry, the distance between two points is the sum of the absolute differences of their coordinates. A taxicab circle with a radius of 5 consists of all points whose coordinates (x, y) satisfy the equation |x| + |y| = 5. To find the number of points on this circle, we can check all possible integer values for x and y that satisfy the equation. By considering the combinations of positive and negative values for x and y, we can count a total of 20 distinct points on the circle.
To determine the only degree measure that could not be the sum of the angle measures of a spherical triangle, we need to understand some properties of spherical triangles. In spherical geometry, the sum of the angles of a triangle is always greater than 180 degrees, and specifically, less than 540 degrees. Therefore, we can eliminate the degree measures 220°, 300°, and 544° as they are all within this range. The only remaining option is 182°, which is the correct answer.