Assignment 3B (Due Wed, 10/22)

Read Chapter 4 in B&B, including the proofs of the theorems. 

Constructing Lengths from Ratios

3.5 Draw a segment AB with length = 10 cm.

a)      Construct with straightedge and compass two points C and D on line AB so that |AC|/|BC| = 4/9 = |AD|/|BD|.  Do this with straightedge and compass and not a marked ruler.  Show your work.

b)      Then, considering a ruler on the line AB with A = 0 and B = 10, measure with your marked ruler and find a good approximate measurement for the numbers corresponding to C and D.

c)      Finally, use algebra to compute exactly (as a rational number, not a decimal at first – then you can give the decimal approximation) the values of the numbers corresponding to C and D.  (If they don’t match up well with your earlier results, then redo something to reconcile the two.)

Trapezoid Midline

3.6 A quadrilateral is a trapezoid if two sides are parallel. Prove that the line joining the midpoints of the other two sides of the trapezoid is parallel to the two parallel sides.  (Hint:  See B&B, p. 116, #24)

Angle Bisector Ratio Theorem

3.7 Let ABC be a triangle.  If S is a point on segment AC so that ray BS is the bisector of angle ABC, then prove that S divides AC in the ratio proportional to the other two sides, i.e., |AS/|SC| = |AB|/|BC|.  (Hint:  See B&B, p. 116, #25)

Concurrent Transversals

3.8 Prove that if 3 or more lines are concurrent, then they cut off proportional segments on two parallel lines.  (Hint:  See B&B, p. 126, #3)

Circumcircle of a right triangle

3.9 Let ABC be a right triangle with angle C = right angle.

• Prove that if M is the midpoint of the hypotenuse AB, then |MA| = |MB| = |MC|. (Note: There are several fairly simple approaches to prove this, but be careful. There are so many relationships present that one must be careful not to assume the conclusion before proving it.)
• Construct such a triangle ABC and construct its circumcircle. Tell what is the circumcenter.

External Tangent -- do NOT turn this in (you did it in lab) but be sure you know how to do it both with Sketchpad and on paper with straightedge and compass

Draw a circle c and a point E outside the circle. Construct with straightedge and compass the two tangent lines to the circle through E.