Question

Complete the coordinate proof of the theorem. Given: ABCD is a rectangle. Prove: The diagonals of ABCD are congruent. Rectangle in quadrant 1 of the coordinate plane with vertices at A(0, 0), B( , 0), C(a, b), and D(0, ). The length of AC is equal to ________. The length of BD is equal to ________. The diagonals of the rectangle have the same length. Therefore, the diagonals of ABCD are congruent.

Answer 1

Using the Pythagorean Theorem, we established that the lengths of diagonals AC and BD in rectangle ABCD are equal, therefore proving that the diagonals are congruent.

Step-by-step explanation:

To prove that the diagonals of rectangle ABCD are congruent, we will use the Pythagorean Theorem and the properties of a rectangle. Rectangle ABCD has vertices at A(0, 0), B(a, 0), C(a, b), and D(0, b). This implies that AB and CD are horizontal lines and AD and BC are vertical lines, since opposite sides of a rectangle are parallel and equal in length.

To find the length of diagonal AC, we see that triangle ABC is a right triangle with legs of length 'a' (AB) and 'b' (BC). By applying the Pythagorean Theorem, the length of AC can be calculated:

AC = √(a² + b²)

Similarly, for diagonal BD, triangle ABD is a right triangle with legs of length 'a' (AB) and 'b' (AD). Thus:

BD = √(a² + b²)

Since both AC and BD have the same length, we conclude that the diagonals of rectangle ABCD are congruent.