MATH 161 SAMPLE FINAL EXAM SOLUTIONS 1. Euclidean geometry
... should think of a way to construct a segment satisfying the definition of “height” so that you can use the hypothesis on the heights.) Problem 4 (4 points). Prove that if f is an isometry and f ◦ f is the identity function, then f has a fixed point. Solution. Assume that f is an isometry and f ◦ f i ...
... should think of a way to construct a segment satisfying the definition of “height” so that you can use the hypothesis on the heights.) Problem 4 (4 points). Prove that if f is an isometry and f ◦ f is the identity function, then f has a fixed point. Solution. Assume that f is an isometry and f ◦ f i ...
Unit 3A: Parallel Lines.docx
... Mathematical skills and understandings are used to solve real-world problems. Problem solvers examine and critique arguments of others to determine validity. Mathematical models can be used to interpret and predict the behavior of real world phenomena. Recognizing the predictable patterns in mathema ...
... Mathematical skills and understandings are used to solve real-world problems. Problem solvers examine and critique arguments of others to determine validity. Mathematical models can be used to interpret and predict the behavior of real world phenomena. Recognizing the predictable patterns in mathema ...
M 1312 6.2 1 Definition: a tangent is a line that intersects a circle at
... Definition: A polygon is inscribed in a circle if its vertices are points on the circle and its sides are chords of the circle. Equivalently, the circle is said to be circumscribed about the polygon. The polygon inscribed in a circle is further described as a cyclic polygon. ...
... Definition: A polygon is inscribed in a circle if its vertices are points on the circle and its sides are chords of the circle. Equivalently, the circle is said to be circumscribed about the polygon. The polygon inscribed in a circle is further described as a cyclic polygon. ...
1 Topic 1 Foundation Engineering A
... A body weighs 40 N in the earth’s gravitational field, but rests on a plane inclined at an angle θ (20◦ ) to the horizontal. If the body is stationary, that is in "static equilibrium", determine the following: a) Resolve the weight f vector (weight is a force different from mass) into a perpendicula ...
... A body weighs 40 N in the earth’s gravitational field, but rests on a plane inclined at an angle θ (20◦ ) to the horizontal. If the body is stationary, that is in "static equilibrium", determine the following: a) Resolve the weight f vector (weight is a force different from mass) into a perpendicula ...
Chaptus Threeus Reviewus (Latin for there is a test on Wednesday
... (Latin for there is a test on Wednesday. Don’t forget about Sketchpad!) Vocabulary: Skew lines, parallel lines, alternate interior angles, same side interior angles, corresponding angles, transversal, regular polygon, equilateral, equiangular Theorems: (These will NOT be given to you on test day) If ...
... (Latin for there is a test on Wednesday. Don’t forget about Sketchpad!) Vocabulary: Skew lines, parallel lines, alternate interior angles, same side interior angles, corresponding angles, transversal, regular polygon, equilateral, equiangular Theorems: (These will NOT be given to you on test day) If ...
Bisect a Line
... given distance Across Corners (Inscribed) Given distance AB across the corners, draw a circle with AB as the diameter With A and B as centers and the same radius, draw arcs to intersect the circle at points C, D, E, and F Connect the points to complete the hexagon ...
... given distance Across Corners (Inscribed) Given distance AB across the corners, draw a circle with AB as the diameter With A and B as centers and the same radius, draw arcs to intersect the circle at points C, D, E, and F Connect the points to complete the hexagon ...
Riemannian connection on a surface
For the classical approach to the geometry of surfaces, see Differential geometry of surfaces.In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form . These concepts were put in their final form using the language of principal bundles only in the 1950s. The classical nineteenth century approach to the differential geometry of surfaces, due in large part to Carl Friedrich Gauss, has been reworked in this modern framework, which provides the natural setting for the classical theory of the moving frame as well as the Riemannian geometry of higher-dimensional Riemannian manifolds. This account is intended as an introduction to the theory of connections.