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Jimmy wrote the following proof to show that the diagonals of rectangle ABCD are congruent:

Proof:

1. In rectangle ABCD, we have opposite sides AB = CD and AD = BC by definition of a rectangle.
2. Diagonals AC and BD can be considered as two sides of two triangles: △ABC and △CDA.
3. By the properties of rectangles, angles ∠ABC and ∠CDA are right angles, making △ABC and △CDA right triangles.
4. By the Pythagorean Theorem, AC = √(AB² + BC²) and BD = √(AD² + DC²).
5. Since AB = DC and AD = BC, we have AC = BD.
6. Therefore, the diagonals AC and BD are congruent.

This completes the proof.

Answer :

Final answer:

The diagonals of a rectangle are congruent. This can be proven using the Pythagorean theorem. By understanding the properties of a rectangle and the concept of congruent triangles, Jimmy's proof is correct.

Explanation:

The provided question describes a proof related to the diagonals of a rectangle. In mathematics, the diagonals of a rectangle are indeed congruent, so Jimmy's proof is correct. To further explain this concept, we can consider the properties of a rectangle and use the Pythagorean theorem to show that the diagonals are of equal length.

First, we need to understand that a rectangle is a quadrilateral with four right angles. This means that the opposite sides of a rectangle are parallel and congruent. When we draw the diagonals of a rectangle, we divide it into four congruent right triangles (two acute triangles and two obtuse triangles).

To prove that the diagonals are congruent, we can consider one of the right triangles formed by a diagonal. Let's call the length of the rectangle's shorter side 'a' and the length of the longer side 'b'. The diagonal divides the rectangle into two congruent right triangles, and we can use the Pythagorean theorem to find the length of the diagonal.

  1. Consider one of the right triangles formed by a diagonal.
  2. Using the Pythagorean theorem, the sum of the squares of the two legs (the lengths of the sides of the rectangle) is equal to the square of the hypotenuse (the diagonal).
  3. Since the opposite sides of a rectangle are congruent, the lengths of the sides are 'a' and 'b'.
  4. By substituting 'a' and 'b' into the Pythagorean theorem, we get a^2 + b^2 = c^2, where 'c' represents the length of the diagonal.
  5. By simplifying the equation and taking the square root of both sides, we find that c = sqrt(a^2 + b^2).
  6. Since 'a' and 'b' are the lengths of the sides of the rectangle, c represents the length of the diagonal.
  7. Therefore, the diagonals of a rectangle are congruent.

Learn more about Congruent diagonals in rectangles here:

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Thank you for reading the article Jimmy wrote the following proof to show that the diagonals of rectangle ABCD are congruent Proof 1 In rectangle ABCD we have opposite sides AB. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!

Rewritten by : Jeany

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