Problem 58 - Precalculus Mathematics for Calculus

Question

Use slopes to show that \(A(-3,-1), B(3,3),\) and \(C(-9,8)\) are vertices of a right triangle.

Step-by-Step Solution

Answer

AB and CA are perpendicular, forming a right triangle at vertex A.

Step 1: Calculate Slope of AB

To find the slope of line AB, use the formula for the slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, the coordinates of A are (-3, -1), and of B are (3, 3). Substitute these into the formula: \( m_{AB} = \frac{3 - (-1)}{3 - (-3)} = \frac{4}{6} = \frac{2}{3} \).

Step 2: Calculate Slope of BC

Use the slope formula for line BC with points B(3, 3) and C(-9, 8). Substitute these into the formula: \( m_{BC} = \frac{8 - 3}{-9 - 3} = \frac{5}{-12} = -\frac{5}{12} \).

Step 3: Calculate Slope of CA

Apply the slope formula to line CA using points C(-9, 8) and A(-3, -1). Substitute these into the formula: \( m_{CA} = \frac{-1 - 8}{-3 - (-9)} = \frac{-9}{6} = -\frac{3}{2} \).

Step 4: Analyze Slopes for Perpendicularity

The triangle is a right triangle if any two sides are perpendicular, which means their slopes are negative reciprocals. For slopes to be negative reciprocals, \( m_1 \times m_2 = -1 \). Check: \( m_{AB} \times m_{CA} = \frac{2}{3} \times -\frac{3}{2} = -1 \). This confirms sides AB and CA are perpendicular.

Key Concepts

Slope Formula

The slope formula is a fundamental mathematical tool used to determine the steepness or inclination of a line on the coordinate plane. The formula is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are coordinates of two distinct points on the line.

• The numerator \( (y_2 - y_1) \) represents the vertical change between the two points, which is often called the "rise".
• The denominator \( (x_2 - x_1) \) is the horizontal change, known as the "run".
• By dividing these two quantities, the slope, or rate of change, is obtained.

Using the slope formula, you can determine how one variable changes with respect to another. This is particularly useful in identifying relationships between points and helps in determining lines that are parallel or perpendicular. Remember, a positive slope indicates an upward incline, while a negative slope shows a downward tilt.

Perpendicular Lines

In geometry, two lines are perpendicular if they intersect at a 90-degree angle. This is a key property often queried in problems involving right triangles.For lines graphed on a coordinate plane, perpendicular lines have slopes that are negative reciprocals of each other. Here’s a quick breakdown:

• When two slopes are negative reciprocals, their product equals \(-1\).
• If the slope of one line is \( m \), the slope of a line perpendicular to it will be \(-\frac{1}{m}\).
• This relationship allows you to quickly assess whether two lines are perpendicular without needing to measure angles directly.

Understanding this concept is crucial for solving problems that involve proving the perpendicularity of lines, especially in triangle problems where you might need to confirm the presence of a right angle.

Negative Reciprocals

Negative reciprocals are numbers that, when multiplied together, result in \(-1\). This concept is vital in understanding the slopes of perpendicular lines.

• To find the negative reciprocal of a number \( m \), you compute \(-\frac{1}{m}\).
• For example, the negative reciprocal of \( \frac{2}{3} \) is \(-\frac{3}{2}\).
• If the product of two numbers is \(-1\), those numbers are considered negative reciprocals of each other.

In the context of geometry and coordinate planes, if the slope of one line segment is the negative reciprocal of another, it implies a perpendicular relationship between the segments. Recognizing this relationship is helpful not only in geometry but also in advanced mathematical fields like calculus and algebra, where understanding the behavior of lines is essential.