GeoMICA - Mrs. Matthews Class
... 10. A point P(7, 4) is translated 3 units down and 1 unit right. The image P′ of P is then rotated 180° clockwise around the origin, resulting in point P″. What are the coordinates of P″? A. (-8, -1) B. (-4, -5) C. (1, -8) D. (5, -4) 11. Parallelogram WXYZ is shown, including diagonal ...
... 10. A point P(7, 4) is translated 3 units down and 1 unit right. The image P′ of P is then rotated 180° clockwise around the origin, resulting in point P″. What are the coordinates of P″? A. (-8, -1) B. (-4, -5) C. (1, -8) D. (5, -4) 11. Parallelogram WXYZ is shown, including diagonal ...
1 Hemmer`s Axioms for Synthetic Euclidean Geometry (Geometry
... For any two distinct points, there is exactly one line incident with both of them. There exists at least two distinct points incident with each line. There exists at least three points incident with each plane, not all three of which are incident with the same line. For any three distinct points tha ...
... For any two distinct points, there is exactly one line incident with both of them. There exists at least two distinct points incident with each line. There exists at least three points incident with each plane, not all three of which are incident with the same line. For any three distinct points tha ...
1 Solution of Homework
... Figure 3: Construction of a triangle with angles of 45◦ , 60◦ and 75◦ . By the converse of Thales’ Theorem, all the vertices F of the right angles with sides through the two points O and P lie on Thales’ circle with diameter OP . 10 Problem 1.4 (Tangents to a circle). Given is a circle C with cente ...
... Figure 3: Construction of a triangle with angles of 45◦ , 60◦ and 75◦ . By the converse of Thales’ Theorem, all the vertices F of the right angles with sides through the two points O and P lie on Thales’ circle with diameter OP . 10 Problem 1.4 (Tangents to a circle). Given is a circle C with cente ...
geometry curriculum - Pompton Lakes School District
... predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if correspon ...
... predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. 7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if correspon ...
Big Ideas - Learn Alberta
... What makes shapes alike and different can be determined by an array of geometric properties. For example, shapes have sides that are parallel, perpendicular or neither; they have line symmetry, rotational symmetry or neither; they are similar, congruent or neither (p. 204). In Van de Walle and Lovin ...
... What makes shapes alike and different can be determined by an array of geometric properties. For example, shapes have sides that are parallel, perpendicular or neither; they have line symmetry, rotational symmetry or neither; they are similar, congruent or neither (p. 204). In Van de Walle and Lovin ...
7.3 Proving Triangles Similar
... Question about homework? • Any questions on the homework? (choose random problems) ...
... Question about homework? • Any questions on the homework? (choose random problems) ...
Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3-dimensional Euclidean space three parameters are required. They can be given in several ways, Euler angles being one of them; see charts on SO(3) for others. Euler angles are also used to describe the orientation of a frame of reference (typically, a coordinate system or basis) relative to another. They are typically denoted as α, β, γ, or φ, θ, ψ.Euler angles represent a sequence of three elemental rotations, i.e. rotations about the axes of a coordinate system. For instance, a first rotation about z by an angle α, a second rotation about x by an angle β, and a last rotation again about z, by an angle γ. These rotations start from a known standard orientation. In physics, this standard initial orientation is typically represented by a motionless (fixed, global, or world) coordinate system; in linear algebra, by a standard basis.Any orientation can be achieved by composing three elemental rotations. The elemental rotations can either occur about the axes of the fixed coordinate system (extrinsic rotations) or about the axes of a rotating coordinate system, which is initially aligned with the fixed one, and modifies its orientation after each elemental rotation (intrinsic rotations). The rotating coordinate system may be imagined to be rigidly attached to a rigid body. In this case, it is sometimes called a local coordinate system. Without considering the possibility of using two different conventions for the definition of the rotation axes (intrinsic or extrinsic), there exist twelve possible sequences of rotation axes, divided in two groups: Proper Euler angles (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y) Tait–Bryan angles (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z). Tait–Bryan angles are also called Cardan angles; nautical angles; heading, elevation, and bank; or yaw, pitch, and roll. Sometimes, both kinds of sequences are called ""Euler angles"". In that case, the sequences of the first group are called proper or classic Euler angles.