## ________________

Sullivan has been making doodles by connecting the midpoints of the sides of a polygon to form a new polygon inside it, trying this process on different starting polygons.

## ________________

The midpoints of the sides of quadrilateral ABCD form the vertices of a rectangle. Angle CAB is 17 degrees.

What is the measure of angle ABD?

### Pent-up-gon

The midpoints of the sides of pentagon ABCDE form the vertices of a regular pentagon.

What is the length of side BC divided by the length of side CD?

## Solutions to week 75

Some Difference. First, let’s find the smallest and largest birthday sums to get an idea of how big the difference could possibly be. We’ll write dates like 68-2-18 for the example birthday, Feb 18, 1968, so that you can just add up the three numbers to get the birthday sum. The smallest possible birthday sum occurs for someone born on 00-1-1, namely 2; we find this by making each of the components as small as possible. Making each of the components as large as possible, the largest birthday sum, 142, goes to someone born on 99-12-31. But notice that 00-1-1 is the day after 99-12-31. So the largest possible difference of birthday sums of people born on consecutive days is 140.