Math 113 Test 3 Review

1. Know the Pythagorean Theorem; be able to state it (in English!). Given the lengths of two sides of a right triangle find the length of the other side. Know how to tell if a triangle is a right triangle. How does the “puzzle proof” with the 4 triangles and the square show that the Pythagorean Theorem is true?

Suppose Bubba is building a wall that is 10 ft. tall and 24 feet long. As usual, his right angle isn't right. He measured one of the diagonals as 26.2 ft. Which diagonal D1 or D2 did he measure? Explain.

If you assume that the angle is right, the length of both diagonals would be √(102 +242)=26 Since the measured one is longer the angle must be obtuse (greater than 90) so its D1.

2. State the Art Gallery Theorem completely accurately and in your own words and know how to apply it. Be able to choose the best-suited points to put the cameras in an art gallery. Be able to divide the museum into triangles and label the vertices. Be able to do problems like these.

Sketch a gallery with 12 vertices which needs 4 cameras

How many cameras will you need for a gallery with 8 vertices? This would require (8/3 rounded down), so 2 cameras. Can you draw a gallery with 12 vertices which needs 5 cameras? There is no such shape. The maximum needed is 12/3 = 4.

4. Sketch a Golden Rectangle. Given any rectangle, determine whether or not it is a Golden Rectangle. What happens if you start with a Golden Rectangle and remove the largest square from it? The remainder is also a golden rectangle. Given the base (long side) of a Golden Rectangle, find its height (short side). To do this divide by φ. Given the height (short side) of a Golden Rectangle, find its base (long side). To do this multiply by φ. Be able to find the area of a Golden Rectangle given either its base or its height. Be able to draw the Golden Spiral and find its center.

A rectangle has sides of length 233 cm by 144 cm. Is this a golden rectangle? Explain how you are determining this. 1+ 5 If it were golden, you would have 233/144 = φ . But since φ= √ , it is an 2 irrational number. Or, if you removed a square from the rectangle and the smaller one was golden you would have 233/144 = 144/89 which is not true. Cross multiple to get 233 ∙ 89 = 144 ∙ 144 and note that the left side is odd and the right side is even. These ratios are close to φ because they are Fibonacci numbers.

5. Understand tessellations, symmetry, rigid symmetry and symmetry of scale. Be able to draw a triangle for the basic building block of the Pinwheel Pattern and draw supertiles of the Pinwheel Pattern from five smaller tiles. Given a diagram of the Pinwheel Pattern, outline a five-tile supertile and a 25-tile supersupertile.

Consider the tessellation below: If the tiling is rotated 120º about point A is this a symmentry? No. The three angles at A are not all equal so it doesn't have 120º symmetry.

If the tiling is moved so that point A moves to B, is this a symmetry? Yes, the two patterns would match each other. If the tiling is moved so that point A moves to C, is this a symmetry? No. But if you added a 180º rotation to the shift, yo would get the same pattern. Are there any symmetries of scale? Explain. No. if you combined 2 hexagons and 2 squares above and below, you would the get a larger hexagon, but there are no larger squares to go with them.

6. Know all about the Platonic solids - their names and various numbers of edges, vertices and faces and their duals.

7. (TR classes) Know what the Euler characteristic of a graph or a solid is and how to compute it.