ABCD and EFGH are squares. If JH = 4 cm and JC = 9 cm, then what is the area of the shaded region?
First, we need to find the length of the sides of the squares. Since JH is a side of square EFGH, we know that the length of the side of square EFGH is 4 cm.
To find the length of the side of square ABCD, we can use the Pythagorean theorem:
JC^2 = JA^2 + AC^2
9^2 = JA^2 + (4+CD)^2
81 = JA^2 + 16 + 8CD + CD^2
65 = JA^2 + 8CD + CD^2
We can also use the fact that the diagonal of square ABCD is equal to JC, which is 9 cm. Using the Pythagorean theorem again:
JC^2 = 2s^2
9^2 = 2s^2
s = (9^2 / 2)^(1/2)
Now we can find CD:
CD = s - 4
CD = (81/2)^(1/2) - 4
To find the area of the shaded region, we need to subtract the area of square EFGH from the area of square ABCD.
Area of square EFGH = (4 cm)^2 = 16 cm^2
Area of square ABCD = s^2 = (81/2) cm^2
Area of shaded region = (81/2) cm^2 - 16 cm^2
Area of shaded region = 49.5 cm^2