Math 621: Elementary Geometry |
The final exam for this class will be cumulative. You should study the questions on the midterm review guide as well as the questions listed below.
1. Prove that if \(d\) is the distance between the centers of two
intersecting circles, \(c\) is the length of their common chord, \(r\) and
\(r'\) their radii, then the circles are orthogonal if and only if
\(cd = 2rr'\).
2. You must be able to construct the image of a triangle under any of
the plane transformations that we studied in class.
3. The \(xy\)-plane, \(\mathbb{R}^2\), is mapped to itself by the
reflection \(R_\ell\) where \(\ell\) is the line with equation \(x - 2y = 4\).
Find the images of the points \((0,0)\), \((1,1)\), \((-3,-2)\), \((4,0)\), and
\((6,-1)\).
4. The \(xy\)-plane, \(\mathbb{R}^2\), is mapped to itself by the
reflection \(R_\ell\) where \(\ell\) is the line with equation \(x - 2y = 0\).
Find the matrix representation of the transformation.
5. Given two non-concentric circles with different radii, find their centers of similitude.
6. Given three non-concentric circles with different radii, whose centers are not collinear,
show that the external center of similitude of one pair of circles in collinear with the internal
centers of similitude of the other two pairs.
7. Given a two isometric triangles in the plane, you must be able to find the lines
that may be used to construct the isometry between the two triangles.
8. You must be able to invert any given plane object (e.g., line, circle, single point)
through a given circle.
9. Consider a square contained entirely in a circle with one vertex at the center of the
circle. Construct the image of the square and its diagonals under inversion in the circle.
10. You must be able to construct lines, triangles, rectangles, and circles in both models
of hyperbolic geometry (the disk and the upper-half plane).
11. You must be able to measure distances in hyperbolic geometry (both models).
12. You must be able to draw a variety of triangles in the extended plane model of elliptic
space, including some with a vertex at \(\infty\).
13. You must be able to measure distances in taxi-cab geometry.
14. You must be able to sketch lines, ellipses and hyperbolas in hyperbolic geometry.
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