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Chapter 9: Problem 14

Write a coordinate proof to show that the diagonals of a rectangle are congruent.

Short Answer

Expert verified
The diagonals of a rectangle are congruent. Their length is equal to \( \sqrt{c^2 + d^2} \), where c and d are the lengths of the sides of the rectangle.

Step by step solution

01

Setting up coordinates

Consider a rectangle ABCD with coordinates A(0,0), B(c,0), C(c,d) and D(0,d) where c and d are the lengths of the sides of the rectangle.
02

Find length of Diagonal AC

To find the length of diagonal AC, apply the distance formula derived from the Pythagorean theorem. This gives \( AC = \sqrt{(c-0)^2 + (d-0)^2} \) which simplifies to \( AC = \sqrt{c^2 + d^2} \).
03

Find length of Diagonal BD

Similarly, apply the distance formula to find the length of diagonal BD. This gives \( BD = \sqrt{(c-0)^2 + (0-d)^2} \) which simplifies to \( BD = \sqrt{c^2 + d^2} \).
04

Conclusion

Since the lengths of the diagonals AC and BD are both \( \sqrt{c^2 + d^2} \), we can conclude based on these calculations that they are equal in length, and therefore congruent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rectangle Properties
A rectangle is a special type of quadrilateral. It has some unique properties that make it quite interesting. One of the key features of a rectangle is that all its interior angles are right angles, measuring 90 degrees each. This ensures that opposite sides are equal and parallel.
Another important property is the symmetry rectangles display. The opposite sides are not only equal but also parallel.
  • All angles are right angles.
  • Opposite sides are equal and parallel.
  • Symmetrical about both diagonals.
This symmetry plays a crucial role in the congruency of diagonals as well. Remembering these properties helps us understand not only the shape itself but its behavior in different mathematical problems.
Congruent Diagonals
Congruency in geometry refers to shapes or lines being identical in form and size. For rectangles, congruent diagonals mean both diagonals are equal in length.
This is significant because the diagonals are lines drawn from one corner of the rectangle to another, splitting the rectangle into two congruent triangles. Thus, each diagonal acts as an axis of symmetry, ensuring the rectangle maintains its form.
In the context of a rectangle with coordinates:
  • The diagonals lie across opposite corners.
  • They divide the rectangle into two equal parts.
  • These diagonals are equal in length by nature of the shape.
This property reinforces the balance and symmetry of a rectangle, making it a fundamental component of its geometrical identity.
Distance Formula
The distance formula is fundamental in coordinate geometry. It's derived from the Pythagorean theorem, providing a way to calculate the distance between two points in a plane. The formula is:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Using this, we can easily find the length of diagonals in a rectangle by substituting the points. For instance, if you have coordinates for points A and C, the formula helps calculate the distance, confirming the diagonals' congruency.
Here's how it works in practice:
  • Identify the coordinates of two points.
  • Apply the formula by plugging in the values.
  • Simplify to find the distance between the points.
This straightforward approach allows us to prove that in a rectangle, diagonals are indeed congruent, showcasing how formulas convert geometric properties into calculable insights.

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Most popular questions from this chapter

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