Problem 14 - Thomas Calculus

Question

Rectangle Find the measures of the angles between the diagonals of the rectangle whose vertices are \(A=(1,0), B=(0,3),\) \(C=(3,4),\) and \(D=(4,1)\)

Step-by-Step Solution

Answer

The angles between the diagonals of the rectangle are 90 degrees each.

Step 1: Verify the Quadrilateral is a Rectangle

To confirm that the given quadrilateral is a rectangle, we need to ensure that opposite sides are equal and the diagonals are equal. The side lengths are:- \(AB = \sqrt{(0-1)^2 + (3-0)^2} = \sqrt{10}\)- \(BC = \sqrt{(3-0)^2 + (4-3)^2} = \sqrt{10}\)- \(CD = \sqrt{(4-3)^2 + (1-4)^2} = \sqrt{10}\)- \(DA = \sqrt{(4-1)^2 + (1-0)^2} = \sqrt{10}\)The diagonals are:- \(AC = \sqrt{(3-1)^2 + (4-0)^2} = \sqrt{20}\)- \(BD = \sqrt{(4-0)^2 + (1-3)^2} = \sqrt{20}\)Since opposite sides are equal and the diagonals are equal, the quadrilateral is a rectangle.

Step 2: Calculate Slope of Diagonals

The slope of diagonal \(AC\) is \( m_1 = \frac{4-0}{3-1} = 2 \).The slope of diagonal \(BD\) is \( m_2 = \frac{1-3}{4-0} = -\frac{1}{2} \).

Step 3: Calculate the Angle Between Diagonals

The angle between two lines with slopes \(m_1\) and \(m_2\) can be calculated using the formula:\[ \tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1m_2} \right| \]Substitute the values of the slopes:\[ \tan(\theta) = \left| \frac{2 - \left(-\frac{1}{2}\right)}{1 + 2 \times \left(-\frac{1}{2}\right)} \right| = \left| \frac{2.5}{0} \right| \]This indicates that the angle between them is a right angle, as vertical and horizontal line differences indicate perpendicular lines.

Step 4: Determine Measures of Angles Between Diagonals

Since the diagonals are perpendicular, the angles between the diagonals are 90 degrees. Rectangles have diagonals that bisect each other at 90 degrees.

Key Concepts

Understanding Angles in a Rectangle

A rectangle is a familiar geomtric shape with four sides. Each angle inside a rectangle is a right angle. That means all the four angles are 90 degrees each. This occurs because the opposite sides of a rectangle are equal in length, and the adjacent sides intersect each other perpendicularly.
When a rectangle is drawn, its definition automatically includes these properties:

• Opposite sides are parallel and equal in length.
• Each interior angle measures 90 degrees.

This geometric property is useful when solving geometry problems because if you are working with any quadrilateral and determine that all interior angles are right angles, then you know it's a rectangle. Recognizing these properties helps solve problems more quickly because rectangles have predictable behavior in geometry.

Diagonals of a Rectangle and Their Properties

Diagonals are crucial lines in geometry. They connect opposite corners of a polygon. In a rectangle, diagonals have unique properties: they are equal in length and bisect each other. This means each diagonal cuts the other into two equal halves.
To calculate the length of a diagonal in a rectangle, you can use the Pythagorean theorem, since each diagonal forms two right triangles. You can find the length, if you know the side lengths of the rectangle with this formula:\[ d = \sqrt{a^2 + b^2} \] where \(a\) and \(b\) are the lengths of two adjacent sides, and \(d\) is the diagonal.
Additionally, in the context of their intersection, the angle between the diagonals can sometimes be crucial information for solving or proving geometric problems. In this particular case, the diagonals of the rectangle are perpendicular, thus forming right angles at their intersection.

Important Geometric Properties of Rectangles

Understanding geometric properties provides a clear understanding of how shapes behave and interact within mathematics. Here are the core geometric properties of rectangles:

• Sides: Opposing sides are equal and parallel.
• Angles: All angles are 90 degrees each, making the rectangle an example of an equiangular quadrilateral.
• Diagonals: Each diagonal is of equal length and they bisect each other at 90 degrees.

Including these properties gives rectangles their regular and predictable form in geometry. Knowing these can greatly assist in solving complex problems that involve rectangles as part of larger diagrams or systems. Recognizing shapes and their inherent properties simplifies many geometric proofs and constructions.