MATH325: Test 2


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.

  1. (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.


  2. (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.


  3. (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.
  4. (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 CA(|AB|) and CB(|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.//

Bonus: (5 points) Given points A, B and C, describe how to construct a line through C which is perpendicular to line AB. For full credit prove that the line you describe is, in fact, perpendicular.