MAT 3050 Geometry

Hard copy due in class Wednesday, April 9

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1. Consider an equilateral triangle in a plane with side length $s$. Assume that this equilateral triangle is a base of a right triangular prism with height $h$.
   1. Use the integration formula from class to derive the formula for the volume of this right triangular prism with equilateral base.
   2. Use your formula to calculate the volume of a right triangular prism with equilateral base of side length 2 and height 3.
2. Consider an equilateral triangle in a plane with side length $s$. Assume that this equilateral triangle is a base of a pyramid with height $h$.
   1. Use the integration formula from class to derive the formula for the volume of this pyramid. (Hint: we’ll learn about different centers of triangles during student presentations; in equilateral triangles, the different centers are actually equal and for this problem you may assume that the “top” of the pyramid projects orthogonally to the center of the triangle).
   2. Use your formula to calculate the volume of a pyramid whose base is an equilateral triangle with side length 4 and height 5.
3. Consider a circle in a plane with radius $r$. Assume that this circle is a base of a right cylinder with height $h$.
   1. Use the integration formula from class to derive the formula for the volume of this right cylinder cylinder. (Hint: you may need to remember that the area of a circle is given by $A=\pi r^2$.)
   2. Use your formula to calculate the volume of a right cylinder with radius 6 and height 7.
4. Consider a circle in a plane with radius $r$. Assume that this circle is a base of a cone with height $h$.
   1. Use the integration formula from class to derive the formula for the volume of this cone. (Hint: for this problem you may assume that the cone point projects orthogonally onto the center of the circle.)
   2. Use your formula to calculate the volume of a cone with radius 8 and height 9.
5. In problems 2 and 4 above for the pyramid and cone, you were told that you could make a simplifying assumption about the positions of the “top” of the pyramid and the cone point. Was this assumption important? How did you use this assumption while deriving the volume formula in each case? If you do not make the simplifying assumption (that is, if you move the point but keep the height the same) does that change the answer? Write a few sentences (or bullet points) answering these questions and explaining your intuition or thinking in detail. Be as specific as possible.

Hard copy due in class Wednesday, April 2

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Hint for Problems 1-4: you should rely on the definitions of the relevant objects for each problem as given in class.

1. Consider a configuration of two lines in 3-space. We have seen two situations where a plane can pass through these two lines. Describe each of these two possible situations (what must the two lines satisfy in order for a plane to pass through them). Be precise and include all details. Then describe a third situation where two lines are configured in such a way that no plane passes through them.
2. Let $P_1$ and $P_2$ be two planes in 3-space that are parallel. Assume that $Q$ is a third plane that intersects $P_1$ in $\ell_1$ and $P_2$ in $\ell_2$. Prove or disprove: $\ell_1$ and $\ell_2$ are parallel in 3-space.
3. Assume that a plane $P$ and a line $\ell_1$ are both perpendicular to another line $\ell_2$. Assume also that $\ell_1$ does not lie in $P$. Prove that $P$ and $\ell_1$ are parallel to one another.
4. Consider two dihedral angles that are vertically opposite to one another. Prove that these dihedral angles are congruent to one another.
5. OpenLab mini-assignment: typing in $\LaTeX$ (instructions here)

Hard copy due in class Wednesday, March 12

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1. Let that the square $\square ABCD$ be similar to the square $\square A’B’C’D’$ with a proportionality constant $\lambda = \frac{5}{2}$ (when going from $\square ABCD$ to $\square A’B’C’D’$). Assume that $| \overline{AB}| = x + 3$ and $|\overline {A’B’}| = 3x + 5$. Determine the perimeter of $\square ABCD$. (Your answer should be a number.)
2. Consider two intersecting chords in a circle. The intersection point cuts one chord into two pieces: one of length $a$ and one of length $b$. The intersection point cuts the other chord into two pieces: one of length $c$ and one of length $d$. Prove that $ab=cd$.
3. Consider the circle in the diagram with center $O$ and radius 6 units. Assume that the arc $\stackrel{\frown}{AB}$ measures $\frac{4 \pi}{3}$ units. Determine the angle $\angle OCB$ in degrees.
   
4. Consider a circle with a chord and a line tangent to the circle at one of the chord endpoints. Let $\theta$ be the (smaller) angle between the chord and the tangent. Let $\phi$ be the central angle formed by the two endpoints of the chord. Prove that $\phi = 2 \theta$.
5. Consider the parallelograms $\square ABCD$ and $\square A’B’C’D’$ in the diagram. Determine the transformation (or sequence of transformations) that takes $\square ABCD$ to $\square A’B’C’D’$.
   • Be explicit about the type of transformation (translation, reflection, rotation) and associated quantities (if a translation, how many units and in which direction? if a reflection, about which line? if a rotation, about which point and how many degrees?) as well as the order of transformations in the sequence.
     
     

Hard copy due in class Wednesday, March 5

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1. Venema 0.9.2 (page 10)
2. Prove that the diagonals of any rhombus are perpendicular to each other.
3. Let the lengths of the diagonals of a rhombus be $d_1$ and $d_2$. Prove that the area of this rhombus is $A = \frac{1}{2} d_1 d_2$.
4. Prove that the diagonals of any isosceles trapezoid are congruent.
5. Prove that the sum of the measures of the exterior angles of any polygon is $360^\circ$.