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 14 inches, and the width qr is labeled as 7 inches. the side lm of the square is labeled as 7 inches.
anna says that the length of diagonal sq is two times the length of diagonal om.
is anna correct? justify your answer and show all your work. your work should state the theorem you used to find the lengths of the diagonals.

You have to use the Pythagoras theorem.
hypotenuse²=leg₁²+leg₂²

We have to calculate the length of diagonal SQ and the length of diagonal OM.

Diagonal SQ:
SR=leg₁=14 in
QR=leg₂=7 in
SQ=hypotenuse

therefore:
hypotenuse²=leg₁²+leg₂²
SQ²=(14 in)²+(7 in)²
SQ²=196 in²+49 in²
SQ²=245 in²
SQ=√(245 in²)
SQ≈15.65 in

Diagonal OM:
LM=leg₁=leg₂=7 in   ( the length of the sides is the same because it is a square)
OM=hypotenuse

therefore:
hypotenuse²=leg₁²+leg₂²
OM²=(7 in)²+(7 in)²
OM²=49 in²+49 in²
OM²=98 in²
OM=√(98 in²)
OM≈9.9 in

Anna says that the length of diagonal SQ is two times the length of diagonal OM, then :
2(OM)=SQ

We multiply the length of diagonal OM by 2:
2(OM)=2(9.9 in)≈19.8 in

We can see that the 19.8 in is not equal to the length of SQ (15.65 in)
19.8 in≠15.65 in
Therefore:

The thought of Anna is wrong, because the length of diagonal SQ (15.65 in) is not two times the length of diagonal OM (19.8 in), 19.8 in≠15.65 in, we have to use the Pythagoras theorem to calculate the length of the diagonals. 

You can find the area of a rectangle by multiplying the length and height. so multiply 1/2*7/4(which is the simpler version of 1 3/4) and you get 7/8 as your answer.