Let's explore how midpoints are found. We have found midpoints with an equation, but how could we also construct a midpoint?

1. What is the definition of a midpoint? 2. First identify the coordinate points of A and B. A = ( ) B = ( ) 3. What is an accurate formula for finding the midpoint of line AB? 4. Use part a and b to find the midpoint. 5. Now use the geogebra tools to find the midpoint. Click on the second icon on the toolbar find the button that says midpoint. Click on point A, then B. Point C should appear with the midpoint. Record the coordinates of point C, by looking at the value of C in the left side toolbar. 6. Can you construct the midpoint without pushing a fancy button? Let's see if we are as smart as Geogebra! a. First we need to delete point C - click on the refresh button in the top right corner of the screen. b. Using the 6th icon in the toolbar (circle), select "Circle with radius and center". Click on either point A or B and enter a value for the radius (hint: it should be a bit larger than what you suspect to be the middle of the segment). Create another circle with center A or B. You should now have two circles, one with center A and one with center B. c. Using the 2nd icon "Point," place a point where the circles intersect; you should create two points. d. Using the 3rd icon "Line," create a line through the two points. e. Place another point, where the original segment AB and the line you just drew intersect. f. You found the midpoint through construction! g. Verify that the point you just created has the same coordinates as you calculated previously.

What is the relationship between chords, arcs, and central angles within a circle?

Use the applet, to see if you can see any relationships between chords and arcs, chords and central angles, as well as arcs and central angles. What do you notice about the relationship between all three of them?

Do you know how Pythagoras really figured out his famous theorem? See if you can answer the questions below, by using applet and your understanding of the Pythagorean Theorem.

1. Using the sliders for the values of a and b, notice that a is one leg of the red triangle and b is the other leg of the red triangle. Also, what is similar about the green, blue, and orange triangle? 2. How do the measures of the measures of the legs relate to the purple squares? What do you notice about the area of the purple square FGJH in relation to the value of a? What do you notice about the area of purple square JKML in relation to the value of b? 3. What would happen if you added the area of square FGJH and JKML? 4. How could you then find the length of one of the sides of the larger purple square ABCD? 5. Measure segment AB to see if your conclusion was correct. 6. What does the Pythagorean Theorem state?