MTH 4/583 Midterm Exam Fall 2017

1. You are welcome to write out your responses on paper or to type your responses in a different program! I really only need a geogebra file through Question 2, and it may be easiest to create a new geogebra file instead of typing into this one. To get the link for a Geogebra file, first choose "Sharing" from the menu. 2. Please name your file as: LastName_FirstName_MTH4583_Winter2017 . When you are finished, save the file a last time and upload a file containing the link (as well as any other links or files) to the midterm folder on the D2L dropbox.

What is the "drag test"? Employ and describe the results of the "drag test" in relation to the figure you constructed above.

Go back to your construction, and now construct the height h of the trapezoid through E (e.g., the perpendicular to TR and PA). Label the height of triangle PEA and the height of triangle TER . What can you say about the ratio in terms of a, b, c, or d?

Write the area of the trapezoid in terms of and .

Use your responses above to explain why triangle PET and triangle EAR must have the same area.

Find a primitive Pythagorean triple with hypotenuse 73. Explain how the geometry of the complex plane can be used to find the primitive triple.

Suppose that x can be written as the sum of two squares, and y can be written as the sum of two squares.  Prove that xy can be also be written as the sum of two squares.  Hint: how can you use complex numbers to write the sum of two squares as a product?

Use polynomial division to find the equation of a parabola that is tangent to the function at and . Form and articulate (at least) two conjectures about relationships between roots, remainders, and tangency behavior.