Reading and other exercises

There will be reading and additional exercises with some material about coordinates, Ceva’s theorem, and related topics. These will be handed out during the week.

 

Practice Problems 1 (Due Wed, 10/28). Do not turn in!

PP 1.1 Distances on rays OA and OB.

Given points O, A, B, suppose that |OA| = a and |OB| = b, and that A' is a point on ray OA with |OA'| = 1/a and B' is a point on ray OB with |OB'| = 1/b.

If |AB| = c, what is |A'B'|?

Your answer should be in terms of a, b, c and you should give a brief but convincing explanation.

 

PP1.2 Constructing a Right Angle (10 points)

Draw on your paper a figure as nearly the same as this one as you can. Construct all points C on line m so that ACB is a right angle. Write briefly the major steps in your construction, especially if you are using tools other than straightedge and compass.

 

 

 

 

Math 444 Assignment 7 (30 points, Due Friday, 10/30)

7.1 Altitudes (20 points)

  1. Construct a triangle ABC and its orthocenter H, then construct the orthocenter of triangle ABH. Where does this appear to be? Prove it.
  2. Extend the class exercise on the angle bisectors of the orthic triangle (done in class Monday) to the case when the triangle ABC is obtuse. Is the same relationship still valid? If not, state a modified version of the relationship which remains valid in all cases.

7.2 Circumcenter of Isosceles (10 points)

Given an isosceles triangle ABC, with |AB| = |AC| = b and |BC| = a, what is the radius of the circumcircle of ABC? (The answer should be in terms of a and b, if possible.)

 

Practice Problems 2 (Friday,10/30). Do not turn in!

PP2.1 A distance relationship

In the figure, the circle has center O and radius r. PQ and NS are diameters. Line P1 Q1 is tangent to the circle at S.

(N, P and P1 are collinear and also N, Q, Q1 are collinear, as they appear to be).

Using Basic Geometry, derive and prove a relationship among the distances |SP1|, |SQ1|, and the radius r.

Note: This figure is related to other geometry that we did this quarter, but this proof should use basic geometry.

 

PP2.2 Distance in a tangent figure

Given a circle with center A and radius r and also a point B exterior to the circle. Let line BC be tangent to the circle at B and segment CD be perpendicular to AB. Find and prove a relationship among AB and AD and the radius r.

 

 

 

PP2.3 Another circle relation

Given a circle with center A and a point B exterior to the circle, let E be a point on the circle so that BE=BA and let F be a point on segment AB so that EF=EA. Find and prove a relationship between AB, AF, and the radius of the circle.

PP2.4 Construct a ratio with straightedge and compass

(a) Draw a segment. Construct with a straightedge and compass a segment which is 3/5 the length of the given segment.

(b) Draw a segment. Construct with a straightedge and compass a segment which is the golden ratio times the length of the given segment. Use this to construct a regular pentagon (not the way in B&B. Start with a side and build from there.

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