Chapter 5 Review, Continued (Day 59)

Today on her Mathematics Calendar 2018, Theoni Pappas writes:

If this circle's equation is x^2 - 10x + y^2 + 4y = 20, what is its radius?

As I've mentioned in the past, I used to consider this to be an Algebra II problem, but the Common Core has pushed it down into Geometry. The U of Chicago text teaches equations for circles in Lesson 11-3, but these don't require completing the square, unlike today's Pappas problem.

In case you need a reminder, the equation for a circle is:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) is the center of the circle and r is the radius. For our given equation:

x^2 - 10x + y^2 + 4y = 20

we complete the square as follows:

x^2 - 10x + 25 + y^2 + 4y + 4 = 20 + 25 + 4
(x - 5)^2 + (y + 2)^2 = 7^2

which tells us that the center is (5, -2) and the radius is 7. So our desired radius is 7 -- and of course, today's date is the seventh.

This is our second review day, which means it's time to look around the web for an activity. The following worksheet comes from Elissa Miller -- yes, I know I haven't linked to Miller's website since August. This link is nearly three years old:

http://misscalculate.blogspot.com/2016/03/geometry-unit-6-quadrilaterals.html

As usual, these don't print well on my computer, so you may wish to go directly to the source. One thing about Miller's quadrilateral worksheets is that some of them mention parallelogram properties that don't appear in the U of Chicago text until Chapter 7. And so instead, I decided to post her trapezoid worksheet instead, even though the other quadrilaterals will appear on tomorrow's test.

Notice that in addition to "isosceles trapezoid," Miller uses the terms "scalene trapezoid" and "right angled trapezoid," neither of which appear in the U of Chicago text. Of course, we can derive their definitions from the analogous terms for triangles.

It also appears that Miller is using the exclusive definition of a trapezoid. For example, Trapezoid Problem #5 asks:

5. AB is not parallel to ____, ____, and ____.

The intended answer is BC, CD, and DA -- clearly AB intersects BC and DA, and since an exclusive trapezoid can't have more than one parallel pair, AB can't be parallel to CD either. But under our inclusive definition, we can't rule out the possibility that AB | | CD. Of course, the way these two segments are drawn, they don't appear anywhere close to parallel at all.