Alana is writing a coordinate proof to show that the diagonals of a rectangle are congruent. she starts by assigning coordinates to a rectangle as shown. then she uses the coordinates to write and simplify expressions for the lengths of the diagonals.

The solution for the problem is:

d (S, P) = √ (0-0) ^2 + √ (b-0) ^2

d (S, P) = √b^2

d (S, P) = b

so we know that, SP = b

d (P, Q) = √ (a-0) ^2 + √ (b-b) ^2

d (P, Q) = √a^2

d (P, Q) = a

so

PQ = a

SQ = c^2 = a^2 + b^2

SQ = √ (a^2 + b^2)

Therefore, the answer is √ (a^2 + b^2).