Final answer:
The diagonals of rectangle DEFG are congruent because both are calculated using the Pythagorean theorem and result in the same length, √ a² + b².
Step-by-step explanation:
To prove that the diagonals of a rectangle are congruent, we can use the coordinates of rectangle DEFG with vertices D(0, b), E(a, b), F(a, 0), and G(0, 0). By applying the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two legs (a and b) is equal to the square of the hypotenuse (c), given by the formula a² + b² = c².
Let's calculate the lengths of the diagonals:
- Diagonal DF: Since D and F are (0, b) and (a, 0), the length of DF is √(0-a)² + (b-0)² = √(-a)² + b² = √ a² + b².
- Diagonal GE: Since G and E are (0, 0) and (a, b), the length of GE is √(0-a)² + (0-b)² = √(-a)² + (-b)² = √ a² + b².
As we see, both diagonals GE and DF have the same length of √ a² + b², hence they are congruent.