Study Sheet for Midterm

What will be on the test?

You should study all the parts of Chapters 1-5 we have covered, plus the concurrence theorems found in GTC ad Chapter 7 of BG, plus the locus theorems, which can be found in BG but which may be emphasized more in GTC (they are also in the locus chapter of B&B).

Review Study Sheet for Quiz 1

Review Study Sheet for Quiz 2

Theorems to Prove (which can be quoted after proving)

• Let ABC be a right triangle with C the right angle. If M, P, and Q are the midpoints of AB, BC, BA, then the segments MP, PQ, QM divide the triangle into 4 congruent subtriangles similar to ABC.

These 3 statements below are equivalent. Proving one will prove the others.

• Given two points A and B, the locus of point C so that ACB is a right angle is the circle with diameter AB (with the points A and B removed).
• A triangle ACB has a right angle at C, if and only if the midpoint M of AB is the circumcenter of the triangle.
• A triangle ACB has a right angle at C, if and only if for the midpoint M of AB, the distances MA = MB = MC.

Locus Theorems

• The locus of points equidistant from A and B is the perpendcular bisector of AB.
• The locus of points equidistant from lines AB and AC is consists of two perpendcular lines, which are the angle bisectors of the 4 angles with vertex A defined by the two lines.
• The locus of points C so that ACB is a right angle is the circle with diameter AB (with A and B removed).
• The locus of points C on one side of line AB so that ACB equals angle t is an arc with endpoint AB and arc angle 360 - 2t.

Thales Theorem

Suppose points A' and B' are on rays OA and OB or else A' and B' are on opposite rays (i.e., O is between AA'and also between BB'.].

• Then A'B' is parallel to AB if and only if
• OA'/OA = OB'/OB if and only if
• angle OAB = angle OA'B' if and only if
• triangle OAB is similar to OA'B'

Note: If these triangles are similar, then also A'B'/AB = OA'/OA = OB'/OB.

Constructions

• Given segments a, b, c, construct a segment x with x/a = b/c.
• For a segment AB and segments c and d, construct points C and D which divide the segment internally and externally in ratio c/d.
• Given a circle c with center O and a point P exterior to the circle, construct the two lines m and n through P which are tangent to c. [If M and N are the intersection points of c with the circle whose diameter is OP, then the lines are MP and NP. You should be able to carry out the construction and/or explain why it works.]
• Given a segment c and a shorter segment a, construct a right triangle with hypotenuse = c and with one leg = a.
• Given two segments a and b, construct a segment c which is the geometric mean of a and b, i.e., c^2 = ab.

From Chapter 5 and Week 5

• Review the concurrence of medians proof from Assignment 4.

Study the review section in the back of each chapter, including chapter 5.

• What is a tangent line to a circle? Why is a line that intersects a circle at one point necessarily perpendicular to the radius at that point? And conversely?
• Inscribed angle theorem and its applications. Include special case of tangents. Think about how this is a more general version of the Carpenter Theorem.
• "If and only if condition" on angles for a quadrilateral to be inscribable in a circle.
• Product relation AE*BE = CE*DE for intersecting chords and intersecting secants. Also a version for tangents.
• Consider examples of quadrilaterals that can be inscribed and circumscribed.

Constructions

• Construct the square root of ab (the geometric mean). This allows the construction of a square with the same area as a rectangle with sides a and b.
• Construct the tangents to a circle through an exterior point P (done in lab 4).
• Construct the common tangents to 2 circles and learn the ratio relationships in the figure.