Homework 3

Complete 5 of the following questions.

Due Wednesday, October 12

1. Complete the following four exercises from Venema, page 56.
   1. 6.4.1
   2. 6.4.2
   3. 6.4.3
   4. 6.4.4
2. Let $d_1$ and $d_2$ be the lengths of the two diagonals of a rhombus. Prove that the area of the rhombus is $A = \frac{1}{2}d_1 d_2$.
3. Let $\alpha$ be an angle in a trapezoid and let $\beta$ be the angle that shares a non-base edge with $\alpha$. Prove that $\alpha$ and $\beta$ are supplementary angles. (If the words are confusing for you: we drew a diagram for this result in class on February 17 that should help; we called this Isosceles Trapezoid Property 2 but we actually don’t need the isosceles assumption.)
4. Prove that the diagonals of an isosceles trapezoid are congruent.
5. Consider two intersecting chords in a circle. The intersection point cuts one chord into two pieces: one of length  and one of length . The intersection point cuts the other chord into two pieces: one of length  and one of length . Prove that .
   • Hint: rewrite this equation so both sides are fractions.
6. Consider a circle with a chord and a line tangent to the circle at one of the chord endpoints. Let  be the (smaller) angle between the chord and the tangent. Let  be the central angle formed by the two endpoints of the chord. Prove that .
7. Show that there is a circle in the plane through three non-collinear points.
   • Hint 1: draw the triangle through the three points and draw the perpendicular bisectors of two of the sides.
   • Hint 2: what you are really proving here is that the circumcircle of a triangle exists (FYI, the circumcircle of a triangle is unique).
8. Assume that the three angle bisectors of a triangle are concurrent (concurrent means they intersect at one point). Prove that the point of concurrency is equidistant from any two sides of the triangle. (This is part (b) of the Angle bisector concurrence theorem from class; a corollary is that the incenter—and hence incircle—of a triangle exists—the incenter/incircle of a triangle are also unique.)

• How did this homework set go for you? Answer the following questions in a few sentences:
• About how long did it take you to complete the homework?
• What was the easiest part for you? What was the hardest part?
• Did you work with anyone else from the class?
• Which outside resources did you use?