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 angle between the diagonals is approximately \(51.34^\circ\).

Step 1: Calculate Diagonals

First, find the equations of the diagonals by using the coordinates of the vertices. Diagonal AC connects points \(A = (1, 0)\) and \(C = (3, 4)\). Diagonal BD connects points \(B = (0, 3)\) and \(D = (4, 1)\).

Step 2: Calculate Diagonal AC Equation

The slope of diagonal AC is \(m_{AC} = \frac{y_2-y_1}{x_2-x_1} = \frac{4-0}{3-1} = 2\). Thus, the equation of line AC in point-slope form is \(y = 2(x - 1)\) or \(y = 2x - 2\).

Step 3: Calculate Diagonal BD Equation

The slope of diagonal BD is \(m_{BD} = \frac{1-3}{4-0} = -\frac{1}{2}\). Thus, the equation of line BD is \(y - 3 = -\frac{1}{2}(x - 0)\) or \(y = -\frac{1}{2}x + 3\).

Step 4: Calculate Angle Between Diagonals

To find the angle between the diagonals, use the formula for the tangent of the angle between two intersecting lines: \(\tan(\theta) = \left|\frac{m_1 - m_2}{1 + m_1m_2}\right|\). Substituting the slopes, \(m_1 = 2\) and \(m_2 = -\frac{1}{2}\), we get \(\tan(\theta) = \left|\frac{2 + \frac{1}{2}}{1 - (2)(-\frac{1}{2})}\right| = \left|\frac{5/2}{2}\right| = \frac{5}{4}\).

Step 5: Calculate the Angle

Convert \(\tan(\theta) = \frac{5}{4}\) to an angle using the inverse tangent function: \(\theta = \arctan(\frac{5}{4})\). Utilizing a calculator, we approximate \(\theta \approx 51.34^\circ\).

Key Concepts

Rectangle Geometry

A rectangle is a quadrilateral, which means it has four sides. In rectangle geometry, there are special properties that simplify calculations. A key feature is that opposite sides are parallel and of equal length, leading to four right angles, each measuring \(90^\circ\). Though the length of the sides can vary, the opposing sides will always hold this equality.

Diagonals in a rectangle connect opposite corners. These diagonals not only bisect each other, but are also always equal in length. This aspect is fundamental when it comes to finding properties like the angle between diagonals. Understanding these basics helps establish how lines are drawn and how they intersect inside the rectangle.

Slope of a Line

The slope of a line gives us a sense of direction and steepness. In simple terms, it represents the change in the vertical direction (y) for a corresponding change in the horizontal direction (x). The formula to calculate slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

• \( m = \frac{y_2-y_1}{x_2-x_1} \)

A positive slope means the line rises as it moves from left to right; a negative slope indicates a fall. A slope of zero signifies a horizontal line, while an undefined slope corresponds to a vertical line.

For our diagonals, diagonal AC has a slope of \(2\), suggesting a steeper upward direction, whereas diagonal BD has a slope of \(-\frac{1}{2}\), indicating a gentle downward slope.

Inverse Tangent Function

The inverse tangent function, also known as the arctangent, is used to find the angle whose tangent is a given number. It is represented as \(\arctan\) or \(\tan^{-1}\). This function is crucial when solving for angles in geometry, particularly in problems involving intersecting lines.

Once you have the tangent value from the slopes of the two lines, \(\tan(\theta) = \frac{5}{4}\) in this context, you can calculate the angle \(\theta\) using:

• \(\theta = \arctan(\frac{5}{4})\)

This calculation gives an angle, providing insights into the orientation of the diagonals in the rectangle. The arctangent function is readily available in most scientific calculators, making it a convenient tool for transforming tangents into angle measures.

Equation of a Line

The equation of a line provides a mathematical way to describe a straight line on a graph. In a cartesian coordinate system, the most commonly used form is the slope-intercept form: \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

For diagonal AC, the equation derived using the slope and a point on the line is:

• \(y = 2x - 2\)

This means the line crosses the y-axis at \(-2\).

Similarly, diagonal BD has its equation as:

• \(y = -\frac{1}{2}x + 3\)

Here, the line crosses the y-axis at \(3\).

Understanding these equations helps in visualizing the lines on the axis, determining their crossing points, and thus calculating the angles between them by examining their interaction.