Midterm Test --- Name___________________________________
Problem 1 (25 points)
In this figure, BC and FE are parallel, and FE/BC = 17/37.
Find each of the following ratios (if this is
possible). Write the ratios as
exact numbers, such as rational numbers, not decimal approximations. Give
brief but convincing reasons. Show your calculations.
Note: You may use any theorems you know.
á What is the ratio AF/AB? Answer ____________
á What is the ratio BD/BE? Answer ____________
á What is the ratio CA/CE? Answer ____________
Problem 2 (30 points)
Prove one of the following theorems (and ONLY one). Circle the proof that you wish to be graded, especially if you try more than one. You can use any theorems you know EXCEPT theorems that are logically equivalent to what you are proving.
a) The three medians of any triangle are concurrent.
b) The three bisectors of the interior angles of any triangle are concurrent.
c) In any right triangle, the midpoint of the hypotenuse is equidistant from each of the 3 vertices.
d) The sum of the interior angles of any triangle is 180 degrees.
Problem 3 (25 points)
Write out one of the proofs of the Pythagorean Theorem (and ONLY one). You can use one of the recent area proofs or an earlier proof.
OR
Prove that if chords AB and CD in a circle intersect at P,
then PA*PB = PC*PD.
Problem 4 -- Construction with choice (20 or 15 points)
DO ONE OF THESE - Do 2A or 2B but not both.
If you try both, cross one out. Construction 2A is worth 15 points. Construction 2B is worth 20 points. (Construction 2A is on this page and 2B on the next page.)
Construction 2A: Tangents through an External Point (15 points)
Construct the lines through A tangent to circle c. Note: The center of c is deliberately left out of the figure.
Construction 2B: Common Tangents (20 points)
Given the two circles, construct two lines that are each tangent to both circles. Note: the centers of the circles are included in the figure.
