MATH325: Test 2

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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.


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  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.


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  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.
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  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.//


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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.