Unit 6: Patterns in Shape
... Quadrilaterals are more complicated than triangles. They have more sides and more angles. In Problem 3, you discovered that using the same four side lengths of a quadrilateral, you could build quite different shapes. Quadrilaterals are classified as convex—as in the case of the quadrilateral below o ...
... Quadrilaterals are more complicated than triangles. They have more sides and more angles. In Problem 3, you discovered that using the same four side lengths of a quadrilateral, you could build quite different shapes. Quadrilaterals are classified as convex—as in the case of the quadrilateral below o ...
MA.912.G.3.3
... Quadrilateral HIJK and quadrilateral LMNO, shown below, both have integer coordinates. Are quadrilateral HIJK and quadrilateral LMNO similar quadrilaterals? A. Yes, both are rectangles. B. No, although the corresponding angles are congruent, the rectangles are not the same size. C. Yes, correspondi ...
... Quadrilateral HIJK and quadrilateral LMNO, shown below, both have integer coordinates. Are quadrilateral HIJK and quadrilateral LMNO similar quadrilaterals? A. Yes, both are rectangles. B. No, although the corresponding angles are congruent, the rectangles are not the same size. C. Yes, correspondi ...
Area of Polygons
... • The base and height of the rectangle are the same as the lengths of the two diagonals of the kite. • The rectangle is divided into 8 triangles; 4 of the triangles fill the kite. For every triangle inside the kite, there is a congruent triangle outside the kite. So, the area of the kite is half of ...
... • The base and height of the rectangle are the same as the lengths of the two diagonals of the kite. • The rectangle is divided into 8 triangles; 4 of the triangles fill the kite. For every triangle inside the kite, there is a congruent triangle outside the kite. So, the area of the kite is half of ...
Shapes Chapter 16 Polyhedra
... often represented by drawings in two dimensions. Without a context suggesting three dimensions, these drawings are ambiguous, and children often do not interpret them in the way they were intended. How to “read” such drawings is important for understanding the ideas they represent, and how to make d ...
... often represented by drawings in two dimensions. Without a context suggesting three dimensions, these drawings are ambiguous, and children often do not interpret them in the way they were intended. How to “read” such drawings is important for understanding the ideas they represent, and how to make d ...
Two-dimensional shapes - Overton Grange Maths KS4
... 7 Triangle ABC is isosceles. The sides AB and AC are both 9.6 cm long. The angles at B and C are both 38°. a Draw a sketch of triangle ABC, showing the lengths of sides AB and AC and the size of the angles at B and C. b Make an accurate drawing of triangle ABC. c Measure the length of BC. Give your ...
... 7 Triangle ABC is isosceles. The sides AB and AC are both 9.6 cm long. The angles at B and C are both 38°. a Draw a sketch of triangle ABC, showing the lengths of sides AB and AC and the size of the angles at B and C. b Make an accurate drawing of triangle ABC. c Measure the length of BC. Give your ...
POLYGONS
... These both have point symmetry because for every point on them, there is another point opposite the centre of symmetry (O) the same distance away. Objects and pictures can often have both rotational and point symmetry. ...
... These both have point symmetry because for every point on them, there is another point opposite the centre of symmetry (O) the same distance away. Objects and pictures can often have both rotational and point symmetry. ...
Graph Theory
... Goal: To color the pre-ordered vertices of G. 1. Assume the vertices of G are ordered v1 , . . . , vn . 2. From i = 1 to n, assign to vi the smallest possible positive integer such that adjacent vertices have distinct integers. 3. In the end the integers represent the different colors used. Definiti ...
... Goal: To color the pre-ordered vertices of G. 1. Assume the vertices of G are ordered v1 , . . . , vn . 2. From i = 1 to n, assign to vi the smallest possible positive integer such that adjacent vertices have distinct integers. 3. In the end the integers represent the different colors used. Definiti ...
Dissections of polygons into convex polygons
... k ≥ 2 then the graph {V, E ∪ {xk x0 }} is called a cycle. A non-empty graph G is called connected if any two of its vertices are linked by a path in G. G is called k-connected if |V | > k and G − X is connected for every set X ⊆ V (G) with |X| < k. The usual way to picture a graph is by drawing a do ...
... k ≥ 2 then the graph {V, E ∪ {xk x0 }} is called a cycle. A non-empty graph G is called connected if any two of its vertices are linked by a path in G. G is called k-connected if |V | > k and G − X is connected for every set X ⊆ V (G) with |X| < k. The usual way to picture a graph is by drawing a do ...
the isoperimetric problem on some singular surfaces
... this open ball, perturb the surface slightly such that the amount of volume enclosed on each side of the singularity changes by a small amount 1v. This volume will be inside the enclosed region on one side of the singularity and outside the enclosed region on the other, so the total change in volume ...
... this open ball, perturb the surface slightly such that the amount of volume enclosed on each side of the singularity changes by a small amount 1v. This volume will be inside the enclosed region on one side of the singularity and outside the enclosed region on the other, so the total change in volume ...
Chapter 9 Geometric Properties (2013)
... 2. At a minimum (to earn a passing grade), your team must find at least 2 items from each section on the vocabulary page. Your group must have at least 5 pictures per person. The more items you find and the higher quality your PowerPoint is, the better your grade will be!! Finding words that no othe ...
... 2. At a minimum (to earn a passing grade), your team must find at least 2 items from each section on the vocabulary page. Your group must have at least 5 pictures per person. The more items you find and the higher quality your PowerPoint is, the better your grade will be!! Finding words that no othe ...
Thrill U. - Dorney Park
... During the winter of 1997, area teachers of physics and mathematics, professionals from Dorney Park, and faculty from Kutztown University gave birth to Thrill U.: The Physics and Math of Thrill Rides. In May 1999, the Thrill U. - Physics Planning Committee presented physics activities in the form of ...
... During the winter of 1997, area teachers of physics and mathematics, professionals from Dorney Park, and faculty from Kutztown University gave birth to Thrill U.: The Physics and Math of Thrill Rides. In May 1999, the Thrill U. - Physics Planning Committee presented physics activities in the form of ...
Lesson 2.4: Angle Properties in Polygons, page 99
... I used software to draw several regular polygons. I noticed that the measure of each interior angle gets closer and closer to 180° as the number of sides increases, so there is less of a “bend” going from one side to another. In other words, the angle is not obvious and it seems to be “smoothed out. ...
... I used software to draw several regular polygons. I noticed that the measure of each interior angle gets closer and closer to 180° as the number of sides increases, so there is less of a “bend” going from one side to another. In other words, the angle is not obvious and it seems to be “smoothed out. ...
Plane Geometry 5.13 Learning Goals: Definitions : Examples:
... Investigate and Describe the results of combining and subdividing plane figures ...
... Investigate and Describe the results of combining and subdividing plane figures ...
chapter-3-understanding-quadrilaterals
... 20) Two adjacent angles of a parallelogram are in the ratio 4:5. Find their measures. 21) The four angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. Find the angles. 22) ABCD is a quadrilateral in which AB = 5 cm, CD = 8 cm and the sum of angle A and angle D is 180°. What is the name of this ...
... 20) Two adjacent angles of a parallelogram are in the ratio 4:5. Find their measures. 21) The four angles of a quadrilateral are in the ratio 3 : 4 : 5 : 6. Find the angles. 22) ABCD is a quadrilateral in which AB = 5 cm, CD = 8 cm and the sum of angle A and angle D is 180°. What is the name of this ...
Polar Representation of Complex Octonions Mehdi JAFARI
... Kansu et al [2]. The complex exponential ei cos i sin generalizes to quaternions by replacing i by any unit quaternion µ since any unit pure quaternion is a root of -1. Hence, any quaternion may be represented in the polar form q q e where is a real angle. As with complex numbers and ...
... Kansu et al [2]. The complex exponential ei cos i sin generalizes to quaternions by replacing i by any unit quaternion µ since any unit pure quaternion is a root of -1. Hence, any quaternion may be represented in the polar form q q e where is a real angle. As with complex numbers and ...
euclidean parallel postulate
... enclosed region. What this suggests is that for piecewise linear figures we will also need to specify what is meant by its interior. The likely formula for the sum of the interior angles of piecewise linear figures can be obtained from Theorem 2.2.1 in conjunction with Sketchpad. In the case of poly ...
... enclosed region. What this suggests is that for piecewise linear figures we will also need to specify what is meant by its interior. The likely formula for the sum of the interior angles of piecewise linear figures can be obtained from Theorem 2.2.1 in conjunction with Sketchpad. In the case of poly ...
Symmetry and Tessellations
... the same way, and that all horizontal pieces look the same as well. Now it becomes an inkblot test. What do you see? Some will think it looks like inter locked men, others may see birds. By decorating each of the newly formed tiles you will create an Escher like tessellation. This method is the simp ...
... the same way, and that all horizontal pieces look the same as well. Now it becomes an inkblot test. What do you see? Some will think it looks like inter locked men, others may see birds. By decorating each of the newly formed tiles you will create an Escher like tessellation. This method is the simp ...
Cauchy`s Theorem and Edge Lengths of Convex
... P apply induction. If all pairs of consecutive angles have sum at most π then 2 α ≤ kπ. Applying condition (1a), 2π(k − 2) ≤ πk, so 2k − 4 ≤ k, so k ≤ 4. Translated to the sphere, Condition (2) seems symmetric with Condition (1) except that it involves arc lengths (corresponding to dihedral angles) ...
... P apply induction. If all pairs of consecutive angles have sum at most π then 2 α ≤ kπ. Applying condition (1a), 2π(k − 2) ≤ πk, so 2k − 4 ≤ k, so k ≤ 4. Translated to the sphere, Condition (2) seems symmetric with Condition (1) except that it involves arc lengths (corresponding to dihedral angles) ...
Origami building blocks: generic and special 4
... pairs of sector angles are equal. Incorporating such constraints changes the combinatorics in determining the possible types of vertices, ultimately resulting in 16 new ones. Furthermore, this allows for the possibility that some of the inequalities that govern the unique and binding folds become eq ...
... pairs of sector angles are equal. Incorporating such constraints changes the combinatorics in determining the possible types of vertices, ultimately resulting in 16 new ones. Furthermore, this allows for the possibility that some of the inequalities that govern the unique and binding folds become eq ...
Complex polytope
In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. On a real line, two points bound a segment. This defines an edge with two bounding vertices. For a real polytope it is not possible to have a third vertex associated with an edge because one of them would then lie between the other two. On the complex line, which may be represented as an Argand diagram, points are not ordered and there is no idea of ""between"", so more than two vertex points may be associated with a given edge. Also, a real polygon has just two sides at each vertex, such that the boundary forms a closed loop. A real polyhedron has two faces at each edge such that the boundary forms a closed surface. A polychoron has two cells at each wall, and so on. These loops and surfaces have no analogy in complex spaces, for example a set of complex lines and points may form a closed chain of connections, but this chain does not bound a polygon. Thus, more than two elements meeting in one place may be allowed.Since bounding does not occur, we cannot think of a complex edge as a line segment, but as the whole line. Similarly, we cannot think of a bounded polygonal face but must accept the whole plane.Thus, a complex polytope may be understood as an arrangement of connected points, lines, planes and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on. Likewise, each line must contain multiple points, each plane multiple lines, and so on.