Assignment 6C (due Wednesday 11/12)

6.4 Multiple ratios in a triangle (from Lab 6)

Let P be a point in a triangle ABC. We denote area of a triangle EFG by <EFG>.

• Define x, y, z as these ratios of areas: x = <PBC>/<ABC>, y = <PCA>/<ABC>, z = <PAB>/<ABC>.

Construct 3 lines through P parallel to the 3 sides of the triangle.  Label the sides as in the figure.

Questions to answer: As usual show your reasoning (briefly)

1. If the side lengths of ABC are a = BC, b = CA, c = AB, then using only x, y, z and a, b, c, write down the lengths of every segment in the figure. You can make a figure and write the lengths in the figure next to the segments.
2. The three shaded triangles are similar to triangle ABC, what is the scaling factor (ratio of similitude) in each case?
3. If the area of triangle ABC is T, what are the areas of the 3 triangles and the 3 quadrilaterals into which ABC is dissected in the figure?  Use algebra to show that the 6 areas add up to the area of ABC.
4. P divides each parallel segment.  Find the values of these 3 ratios, using x, y, z, a, b, c:
• PC2/PB1 = _________
• PA2/PC1 = _________
• PA1/PB2 = _________
5. Let the lines AP, BP, CP intersect the opposite sides of the triangle ABC in points A', B', C'.  Find the ratios
• A'B/A'C = _________
• B'C/B'A = _________
• C'A/C'B = _________
6. For what points in the triangle is x = 0?
7. For what points in the triangle is y = z?
8. If x = y = z, where is P?

6.5 Similar triangles with shared angle - the whole story

In each questions below, A' is a point on ray OA and B' is a point on ray OB. Show your reasoning.

1. Suppose OA = a and OB = b, and triangle OA'B' is similar to triangle OAB with scaling ratio from OAB to OA'B' = K. What is OA'? What is OB'?
2. Suppose OA = a and OB = b, and triangle OB'A' is similar to triangle OAB with scaling ratio from OAB to OB'A' = L. What is OA'? What is OB'?
3. Suppose OA = a and OB = b, and triangle OA'B' is similar to triangle OAB. If OA' = c, what is the scaling ratio from OAB to OA'B? What is OB'?
4. Suppose OA = a and OB = b, and triangle OB'A' is similar to triangle OAB. If OA' = c, what is the scaling ratio from OAB to OB'A' ? What is OB'?

6.6 More tetrahedral relationships

Model: Boxed tetrahedron (models should be cardboard)

• Build a cube, with sides of 3 inches and with one side that opens as a box.
• Draw segments on the cube connecting 4 of the vertices, so that the segments you draw form a regular tetrahedron.
• Build a cardboard regular tetrahedron that fits precisely inside the box, matching the segments you drew.
  

  Volumes

1. Given the tetrahedron in the cube, as in the model, the rest of the cube consists of a number of corner pieces. What is the shape of each piece and what is its shape? What is the volume of each piece?
2. What is the volume of the tetrahedron? What is the ratio of volumes of the tetrahedron and the cube?
3. The midpoints of the edges of the tetrahedron form the vertices of an octahedron (the one met in class with the straw tetrahedra). Where are these vertices located on the cube? Describe the vertices of this octahedron in the cube without reference to the tetrahedron.
4. What is the volume of this octahedron? What is the ratio of volumes of the octahedron and the cube?
   

6.7 Ice Cream Cone Problem

Suppose you have an ice cream cone (a cone with a circular base). The height of the cone is H and the cone holds 100 cubic centimeters of ice cream. If you want to fill the cone partially with 50 cc of ice cream, how high (deep) will be the ice cream in the cone.

Comment. Being an ice cream cone, we put the vertex at the bottom and measure height up from there.