Both figures shown here are squares with a side length of . Notice that the first figure is divided into two squares and two rectangles. The second figure is divided into a square and four right triangles with legs of lengths and . Let’s call the hypotenuse of these triangles .

First of two squares of the same area. This square is divided into the following: A square with side lengths “a”. Two rectangles with side lengths “a” and “b”. A square with side lengths “b”.

Second of two squares of the same area. This square is divided into the following: Four identical triangles on each corner of the square with sides labeled “a” and “b”. A square in the center with unlabeled side lengths.

1. What is the total area of each figure?
2. Find the area of each of the 9 smaller regions shown in the figures and label them.
3. Add up the area of the 4 regions in Figure F and set this expression equal to the sum of the areas of the 5 regions in Figure G. If you rewrite this equation using as few terms as possible, what do you have?

Your teacher will give your group a sheet with 4 figures. Cut out the 5 shapes in Figure 1.

1. Arrange the 5 cut out shapes to fit inside Figure 2. 

2. Now arrange the shapes to fit inside Figure 3. 

3. Check to see that Figure 3 is congruent to the large square in Figure 4.

4. Check to see that the 5 cut out shapes fit inside the two smaller squares in Figure 4.

5. If the right triangle in Figure 4 has legs and and hypotenuse , what have you just demonstrated to be true?

1. Find .

   

   
    
2. Find .

   

3. A right triangle has sides of length 2.4 cm and 6.5 cm. What is the length of the hypotenuse?
4. A right triangle has a side of length and a hypotenuse of length . What is the length of the other side?

5. Find the value of in the figure.