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Private: Learning Math: Geometry
Private: What Is Geometry? Homework
Session 1, Homework
Problem H1
Draw five quadrilaterals, each on its own piece of patty paper. Use one quadrilateral for each construction below.
- Construct the eight medians of the first quadrilateral. (There will be two medians at each vertex.)
- Construct the four midlines of the second quadrilateral.
- Construct the four angle bisectors of the third quadrilateral.
- Construct the four perpendicular bisectors of the sides of the fourth quadrilateral.
- Construct four altitudes in the fifth quadrilateral — one from each vertex.
Problem H2
Try Problem H1 with a few different quadrilaterals. Don’t just use “special” cases, like squares and rectangles. Record any observations about the constructions above. How is the situation different from what you saw with triangles?
Problem H3
Imagine a square casting a shadow on a flat floor or wall. Can the shadow be non-square? Non-rectangular? That is, can the angles in the shadow ever vary from 90°?
TAKE IT FURTHER
Problem H4
Which of these shapes could cast a square shadow, and which could not? Explain how you decided.
Problems H3 and H4 taken from Connected Geometry, developed by Educational Development Center, Inc. pp. 12-13. © 2000 Glencoe/McGraw-Hill. Used with permission. www.glencoe.com/sec/math
Solutions
Problem H1
- Pick a vertex, and then find midpoints of the two sides opposite the chosen vertex. Draw a line segment between the vertex and each of the midpoints. Repeat for the other three vertices.
- Find the midpoints of the four sides. Connect the consecutive midpoints to form the four midlines.
- Pick an angle (vertex). Fold the paper along the line containing the vertex and such that the two sides emanating from the vertex overlap. The crease created bisects the angle. Repeat for the other three angles (vertices).
- Pick a side. Fold the paper so that the endpoints of the side overlap. The crease created defines the perpendicular bisector of the chosen side. Repeat for the other three sides.
- If the quadrilateral is a rectangle, you are done, since its sides are its altitudes. If not, extend its sides in case those lines are needed. Pick a vertex and a side (or extended side) opposite it. Make a fold along a line that contains the vertex such that the two parts of the (extended) side opposite it overlap. Repeat for the other three vertices.
Problem H2
For some quadrilaterals (specifically those which can be inscribed in a circle), the concurrency of perpendicular bisectors holds. For all quadrilaterals, the midlines come in pairs that are parallel. For some quadrilaterals (specifically those which can have a circle inscribed in them), the angle bisectors are concurrent.
Problem H3
A shadow of a square can be a non-square. It can also be a non-rectangle. Yes, angles in a shadow of a square can be different from 90°.
Problem H4
Shapes 1, 2, 6, 7, and 8 can cast a square shadow. One way to visualize this is to think of a non-right square pyramid. Think of a light source as being at the top vertex of the pyramid, and the edges that emanate from it as rays of light. Therefore, any cross section of the pyramid created by a plane that does not intersect the plane containing the base can cast a square shadow (the base). The shapes that might be formed by these cross sections must have four sides, because the pyramid has four sides and must not have an interior angle greater than 180°.
Suggested Reading
Education Development Center, Inc. (2000). Perspective: What Is Geometry? In Connected Geometry: A Habits of Mind Approach to Geometry.Glencoe/McGraw-Hill.
Reproduced with permission from the publisher. Copyright © 2000 by Glencoe/McGraw-Hill. All rights reserved.
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Perspective: What Is Geometry?
