Reminder -- Quiz 1: Friday, 10/17

The quiz will take all or most of the class period. It will include questions, constructions, and problems based on work through 10/15.

• Theorems and problems from Chapters 1, 2, 3 in B&B, also pp. 105-111
• Straightedge constructions from B&B, pp. 173-183 and from assignments
• Constructions and theorems from lab and GTC
• Notes and class activities documented on the class web site.
• Problems from Assignments through 3A.

Reading: Practice the ratio problems, B&B pp. 64-65. Study the rest of Chapter 3 in B&B, especially starred problems, all Principles and Corollaries, and chapter Summary.

In a triangle, an altitude is a line through a vertex perpendicular to the opposite side. The foot of the altitude is the intersection of the altitude with the opposite side (extended to a line if necessary).

Triangle ABC has a right angle at C, with |AB| = c, |BC| = a, |CA| = b. Let D be the foot of the altitude through C.

• Find |CD|, |AD|, BD|, each as an expression in a, b, c. Prove that your answer is correct.

In triangle ABC, let A', B', C' be the feet of the altitudes through A, B, C, respectively.

• Construct two figures that are examples of this, one with all acute angles and one with an obtuse angle.
• Use similar triangles to prove that |AA'||BC| = |BB'||CA| = |CC'||AB|. Be sure your proof covers all cases. Hint: Write one of these equalities as an equation of ratios.

Remark: You may notice a connection of this result with area. This connection will be explored soon, but at the moment we do not (officially) know anything about area; but we do know a lot about similar triangles.

Let ABC be a triangle and let B' be a point on AB and C' a point on AC with |AB'|/|AB| = |AC'|/|AC|.

• If M is the midpoint of BC and N is the intersection of line AM with line B'C', prove that N is the midpoint of B'C'.

Let ABC be an isosceles triangle with AB = AC. Let D be a point on AB with CD = CB. If |AB| = 63 and |BC| = 17, what is |BD|? Show your reasoning.