Exercise 88 Page 48

Practice makes perfect

a The numbers inside the rectangle represent the area of the smaller rectangles, while each number along the side represents the length of that portion of the side.

We can start by filling the small rectangle at the bottom left corner. Since its sides are 11 and 4, its area will be A=11*4= 44. Furthermore, given its sides we can identify the other rectangle's corresponding sides, which should be equal.

To find the other side, we know that the top left smaller rectangle can be decomposed from 12 to 3* 4, since we already know one of its sides is 4. The other side must be 3. We can use this information to label the corresponding sides for other smaller rectangles.

Finally, we find the other side of the top right smaller rectangle if we decompose 39 as 3* 13, since we already know one of its sides is 3. The other side must be 13. We can use this information to label the corresponding side for the lower right smaller rectangle. We can see its area should be given by A= 11* 13 = 143.

Having found all inner rectangles areas, we can add them to get the total area.

As we can see from the picture above, the total area is 238 squared units.

b We can start by matching corresponding sides to the ones we already know.

Now we can decompose 40 as 8* 5, since one of the sides of the bottom left rectangle is 8. Therefore, the other should be 5. Similarly, for the rectangle with the area of 18 we can decompose 18 as 6 * 3. This allows us to find the missing side to be 3. We can also use the new sides to label corresponding sides for the other inner rectangles.

We can now multiply the sides of the top left rectangle and the bottom right one to find their areas. For the top left rectangle we have A=8* 3 =24, and for the bottom right we have A=6 * 5 =30.