Page 35


Problems 1-8


These problems were discussed last semester. In fact, they made up the content of one of your quizzes.



Problem 9


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Suppose Jackie starts at point A and walks counter-clockwise around the figure. When Jackie gets to each vertex, she turns left so many degrees. Answer these questions: (1) When you take the number of "turning" degrees and add them to the angle on the inside of the polygon, what is the sum? (2) How many of the angle pairs are there around this figure? (3) What is the sum of the interior angles? (4) What is the difference of these two sums?



Problem 10


Follow the same reasoning in Problem 9. You should come to the same conclusion AND the same answer!



Problem 11


The SENTRY THEOREM says that the sum of the EXTERIOR ANGLES of polygon are always....?



Problem 12


If the hypotenuse is twice as long as the shorter leg, then by the Pythagorean Theorem, letting s be the shorter leg, the other leg must be



The perimeter of this triangle is given by the expression



Therefore, the shortest side is




Find the lengths of the remaining sides accordingly.



Problem 13


This is a SPACE DIAGONAL question. From the looks of it, I would say no.

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Problem 14


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Page 36


Problem 1


This problem seems to be about THREE IDEAS: (1) The SSA Criterion; (2) Using the distance formula; and (3) recognizing special right triangles.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Make sure you know where each things comes from

(a) This is all about the distance formula and using the distance formula.


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(b) You can certainly use the Distance Formula again, but recognizing that this is a 45-45-90 triangle, we can deduce that the coordinates of C are (4,4).

(c) Distance formula again. I think I will just modify my Nspire document. Be careful. Angle CAB is 45 degrees, so this will limit your answer to just one of the two choices.

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Problem 2


(a) In a cube, each edge has the same length. Let each edge be s. Then each diagonal has a length



(b) If I assign coordinate to the vertices of the cube, one vertex could be (0,0,0) and the opposite vertex would be (s,s,s). The midpoint of the the space diagonal between these two points is



Find the coordinates of the other six vertices, and show that their midpoints are all this point.

(c) Again, using the coordinates as in part (b), you want to select any two of the space diagonals, express them as vectors, and then show their dot product is zero.



Problem 3


The midpoint is just the average of the coordinates.





Problem 4


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Problem 5 and 6


Play with this applet to get a feel for just one possible way to do this.

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All the triangles are SIMILAR, just scaled copies of the original.



Problem 7


In the applet below, try your hand at typing in different combinations of u and v. Remember, the vector will start at the origin, so it wee? -MGP be PARALLEL and THE SAME LENGTH as your desired vector.

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Problem 8


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Problem 9


You should use the POINT TOOL below to plot the points that satisfy the conditions of the problem. Then you should derive the equation of the parabola, simplifying as much as you can, then entering the equation in the input line to see if the equation passes through your points.

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Problem 10


Ahhh...a COUNTING question. If you notice in the picture below, it is possible to draw eight different right triangles that have edge EG as a side. Finish counting.

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Problem 11


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Problem 12


According to the SENTRY THEOREM, what do the exterior angles add up to? In a regular polygon, what is true about the exterior angles?



Problem 13


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Problem 14


Try to construct these pentagons using the rotation tool. Only one will work. Now tesselating is a different question.