MAT 3050 Geometry

Hard copy due in class Wednesday, May 14

Homework #7 is optional: if you turn in Homework #7, your homework average will be determined using all 7 homework sets; if you do not turn in Homework #7, your homework average will be determined using the first 6 homework sets.

See homework guidelines and tips here

1. (From Gele’s presentation) Venema 2.2.4
   • Hint: Sherly proved a relevant result on page 8, so you may use it without proof.
2. (From MS’s presenation) Venema 4.3.4
   • Hint: MS solved Venema 4.2.5 during her presentation; 4.3.4 is similar.
   • MS’s GeoGebra worksheets
3. (From ZJ’s presentation)
   • (a) Venema 5.2.5 (use the diagram on the left side of Figure 5.3 on page 49)
     • Venema’s hint says to use the SAS similarity criterion, which is on page 8; you can use this without proving it
   • (b) With the labels of Figure 5.3, prove that $\mu(\angle B’C’A’) = 180^\circ – 2 \mu(\angle B’CA’)$.
     • Hint: use part (a).
4. (From Sherly’s presentation) Consider Figure 7.1 on page 58 of Venema. Prove that $\square DEFB$ is a parallelogram.
   • By the way, this is the first part of exercise 7.1.5. The rest of the exercise walks you through the proof that $C’$, which is a foot of an altitude, is on the circle determined by the midpoints $D, E,$ and $F$ of the sides of $\Delta ABC$. It generalizes to show that the feet of the altitudes are on the circle determined by the edges’ midpoints, which gives the first 6 points of the 9-point circle!
5. (From Rachel’s and Taspia’s presentations) Venema 8.5.4
   • Hint: Venema probably wants us to use the trigonometric form of Ceva’s theorem (which we didn’t cover), but we can use the traditional form of Ceva’s theorem instead. An ingredient in your proof might be the angle bisector theorem, which states: Let $\Delta ABC$ be a triangle and let $D$ be the point. where the bisector of $\angle CAB$ intersects $\overline{BC}$. Then $\frac{BD}{DC} = \frac{AB}{AC}$. You may use the angle bisector theorem without proving it, but you might like to see this proof on Wikipedia.
6. (Think about this riddle to prepare for Ting’s presentation on spherical geometry) Assume that you walk 1 mile south and see a bear. Then you walk 1 mile west and then 1 mile north and you are at your starting point. What color was the bear?
7. (Make this model to bring to Mariola’s presentation on hyperbolic geometry)
   • Recall that we can build a tetrahedron out of equliateral triangles where 3 triangles meet at a vertex. We can build an octahedron out of equliateral triangles where 4 triangles meet at a vertex. And we can build an icosahedron out of equliateral triangles where 5 triangles meet at a vertex. We saw that if 6 triangles meet at a vertex, then we are building the flat Euclidean plane and not a convex polyhedron.
   • For this exercise, you will build a model of the space where 7 equilateral triangles meet at a vertex.
   • Cut about 20 congruent equliateral triangles out of paper and tape their edges together so that 7 triangles meet at every vertex. The more triangles in your model, the better!
   • If you don’t want to measure the triangles yourself, this site will generate a triangle grid for you that you can print so you can cut out the triangles easily.

Hard copy due in class Wednesday, April 9

See homework guidelines and tips here

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

See homework guidelines and tips here

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)