Problem 1

The points P1, P2, ... , P9 are equally spaced on the circle and divide the circle into 9 congruent arcs. The lines AB, BC, and CA are tangent to the circle at the points P5, P8, and P1, respectively, as shown in the figure.

Write the angle measure in degrees of the vertex angles of each of the following triangles.

Write the angles. No reasoning needed. Triangle P1P5P8: Angle P1 = ______ Angle P5 = ______ Angle P8 = ______ Triangle P8 C P1: Angle P8 = ______ Angle C = ______ Angle P1 = ______

Problem 2

ADBC is a quadrilateral. The diagonals AB and CD intersect at E.

The lengths of some segments are AE = 6, BE = 8, CE = 12, DE = 4, AD = 9 and AC = some number k. THE FIGURE IS NOT TO SCALE!

If it can be determined, compute the length of BC as an exact number or else an expression using numbers and k. Show your reasoning (briefly).

If it can be determined, compute the length of BD as an exact number or else an expression using numbers and k. Show your reasoning (briefly).

Problem 3

Given angle ABC, prove that a point P interior to the angle is equidistant from the rays BA and BC if and only if P is on the angle bisector of ABC.

Note: You can use any theorems except versions of the one you are proving.

Problem 4

Given the segment AB, construct a point C on the segment that divides it in the ratio 1/sqrt 2, i.e., AC/CB = 1/sqrt 2.

Briefly write a few short statements and/or label to make clear what you did.

Problem 5

Given this angle and the point A on one side of the angle, construct a circle c through A so that c is inscribed in the angle, i.e., tangent to both rays that are the sides of the angle.

Briefly write a few short statements and/or label to make clear what you did.