You may use pencil, pens, ruler, compass, protractor and scissors on this test. You may not use a calculator, book or notes. Please take your time on each problem and show your work in the space provided. If you get stuck on a problem, move on to the next and return when you have time.

- (25 points) In the figure shown below, |AB'|/|AB| = 3/4 and line B'C' is parallel to line BC.
What is the exact value of the ratio |B'C'|/|BC|? Justify your answer.

**Name:**

- (25 points) Points A, B, C and D lie on a circle. Segments AD and BC intersect at point P.
Prove triangle ABP is similar to triangle CDP.

**Name:**

- (25 points) Give a list of steps that will construct an equilateral triangle. You need not
prove the triangle is equilateral, but please indicate which vertices and edges make up the triangle.

**Name:**

- (25 points) Find as many errors as you can in the following proof. For each error you find,
describe how you know the statement is incorrect.
**Lemma 3.3.2:**We can construct the perpendicular bisector of any arbitrary line segment AB.**Proof:**Construct C_{A}(|AB|) and C_{B}(|AB|); construct the two points of intersection D and D' of these circles.Construct the intersection of line ST with segment AB; call this intersection point M.

Angle AMS and angle BMS are congruent, complementary angles; therefore they must be right angles.

Angle ASM and angle BSM are congruent because ray ST is an angle bisector.

Segment SM is congruent to itself.

By Angle-Side-Angle, triangle AMS is congruent to triangle BMS.

Segment AM is congruent to segment BM because corresponding parts of congruent triangles are congruent.

Therefore, S must be the midpoint of segment AB.//