Math 444 Midterm
Do all Problems, except for choice on the last constructions 2A and 2B.
Problem 1 (25 points)
Let ABC be a triangle, with M the midpoint of AB. Prove that M is equidistant from A, B and C if and only if angle C is a right angle.
Instructions: Write clearly but succinctly. You may assume all that we know about similar and congruent triangles, perpendicular and angle bisectors, and parallel lines. You may NOT assume theorems about angles inscribed in circles.
Problem 2 (20 points)
The triangle ABC is a right triangle with right angle at C. D is a point on AB so that CD is an altitude. Denote the lengths of the sides of ABC by a = BC, b = CA, c = AB.
Note: In the questions below, you can assume the standard theorems except NOT the Pythagorean Theorem.
(a) Find the length of AD as an expression (i.e., a formula) that involves only a, b, c and no other lengths. Give brief but convincing reasoning.
(b) Find the length of BD as an expression (i.e., a formula) that involves only a, b, c and no other lengths. Give brief but convincing reasoning.
(c) Use your answers above to give a proof of the Pythagorean Theorem.
Problem 3 (5 points)
In a circle, AB and CD are chords that intersect at a point E inside the circle.
State BUT DO NOT PROVE a relationship among the lengths EA, EB, EC, ED.
Problem 4 (15 points)
In this figure, AE/AC = AF/AB = 17/35.
· Tell what is the ratio BD/BE. The answer should be a number expressed as an exact rational number (fraction), not a decimal.
· Give brief but convincing reasons. Show your calculations.
Note: Use any theorems you know.
BD/BE = ____________
Show work here:
In constructions, write a BRIEF listing of KEY steps. No reasons needed. Just make it clear what you did.
Construction 1: Tangent circles (15 points)
The lines AB and CD are parallel. Construct all circles tangent to all 3 lines in the figure.
Construction 2 (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.
