Look at the rectangle and the square:
A rectangle PQRS and square LMNO are drawn side by side. The length SR of the rectangle is labeled as 12 inches, and the width QR is labeled as 6 inches. The side LM of the square is labeled as 6 inches
Sam says that the length of diagonal SQ is two times the length of diagonal OM.
Is Sam correct? Justify your answer and show all your work. Your work should state the theorem you used to find the lengths of the diagonals. (10 points)
To determine if Sam is correct, we can compare the lengths of the diagonals SQ and OM.
First, let's find the length of diagonal SQ in the rectangle PQRS. We can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (in this case, SQ) is equal to the sum of the squares of the lengths of the other two sides.
In rectangle PQRS, we can see that SQ is the hypotenuse of a right triangle with sides QR and SR. So, we have:
SQ^2 = QR^2 + SR^2
Plugging in the values given:
SQ^2 = 6^2 + 12^2
SQ^2 = 36 + 144
SQ^2 = 180
Taking the square root of both sides:
SQ = √180
SQ ≈ 13.42 inches
Now, let's find the length of diagonal OM in the square LMNO. Since LMNO is a square, all sides are equal, including the diagonals. Thus, we can directly find the length of OM:
OM = LM = 6 inches
So, according to our measurements:
SQ ≈ 13.42 inches
OM = 6 inches
Since SQ is not two times the length of OM, Sam is incorrect. Therefore, SQ is not two times the length of OM.
In summary, Sam's statement is incorrect. The length of diagonal SQ in the rectangle is approximately 13.42 inches, while the length of diagonal OM in the square is 6 inches.