Warm-up Question #1:

1) Look at the first circle (upper left), which has a rectangle above the diameter. Are all 4 corners touching the circle? Which corners are touching the circle?

If the rectangle is made longer by stretching it to the right and to the left, so that the bottom corners touch the circle, will the top corners still touch the circle?

If the bottom right corner of the rectangle is stretched to the right so that it touches the circle, and the bottom left corner is stretched to the left so that it touches the circle, and the top corners stay where they are, will the figure still have four 90° angles? Will it still be a rectangle? What kind of figure would it become?

Apply these same questions to the second circle, which has a rectangle below the diameter.

Warm-up Question #2:

2) Look at the third circle, which has a rectangle that is mostly below the diameter. Are all 4 corners touching the circle?

Which corners are touching the circle?

If the rectangle is made longer by stretching it to the right and to the left, so that the top corners touch the circle, will the bottom corners still touch the circle?

If the top right corner of the rectangle is stretched to the right so that it touches the circle, and the top left corner is stretched to the left so that it touches the circle, and the bottom corners stay where they are, will the figure still have four 90° angles? Will it still be a rectangle? What kind of figure would it become?

If the top of the rectangle is stretched upward, until the top corners touch the circle, will the figure still have four 90° angles? Will it still be a rectangle? Will most of the rectangle still be below the diameter? Would the diameter appear to cut the rectangle into 2 equal parts? Would the diameter cut through the middle of the rectangle?

The main question:

3) In Figure 1, which has a rectangle inscribed in a circle, is Line AB the same length as Line CD? How do you know? Is Line BC the same length as Line DA?

Is Arc AB the same length as Arc CD? How do you know? Is Arc BC the same length as Arc DA?

If you add Line AB to Line BC, would the combined length be equal to the combined length of Lines CD and DA?

If you add Arc AB to Arc BC, would the combined length be equal to the combined length of Arcs CD and DA?

Is Arc AC (ABC) the same length as Arc CA (CDA)? Does Line AC cut the circle into 2 equal parts? Is Line AC a diameter of the circle? Does Line AC pass through the center of the circle?

Is Arc BD (BCD) the same length as Arc DB (DAB)? Does Line BD cut the circle into 2 equal parts? Is Line BD a diameter of the circle? Does Line BD pass through the center of the circle?

Is Point E located on Line AC? Is it also located on Line BD? Are both lines diameters? Do both lines pass through the center of the circle? Is Point E the exact center of the circle? How do you know?

4) Look at Figure 2 and ask the same questions as in (3) above. Do the same for all the other figures.

Are all the rectangles the same size? Is Line AB the same length in Figure 9 as in Figure 1? Line BC the same length in Figure 9 as in Figure 1?

The circle in Figure 9 is the same size as the circle in Figure 1. Are the diagonals in Figure 9 the same size as the diagonals in Figure 1? Are all the diagonals in all the figures the same length?

Do all the diagonals pass through the center of the circles?

Are all the diagonals the same length as the diameter?