Problem 58 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

AB and AQ are perpendicular lines, confirming a right triangle.

Step 1: Calculate the Slope of AB

The slope of line segment AB is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For points \(A(-3, -1)\) and \(B(3, 3)\), we have:\[m_{AB} = \frac{3 - (-1)}{3 - (-3)} = \frac{4}{6} = \frac{2}{3}\]

Step 2: Calculate the Slope of BQ

Using the same slope formula, for points \(B(3, 3)\) and \(Q(-9, 8)\), we calculate:\[m_{BQ} = \frac{8 - 3}{-9 - 3} = \frac{5}{-12} = -\frac{5}{12}\]

Step 3: Calculate the Slope of AQ

Again using the slope formula, with points \(A(-3, -1)\) and \(Q(-9, 8)\):\[m_{AQ} = \frac{8 - (-1)}{-9 - (-3)} = \frac{9}{-6} = -\frac{3}{2}\]

Step 4: Check for Perpendicular Slopes

For the triangle to be a right triangle, one pair of slopes must be negative reciprocals, indicating perpendicularity. We compare \( m_{AB} = \frac{2}{3} \) and \( m_{AQ} = -\frac{3}{2} \):\[m_{AB} \times m_{AQ} = \frac{2}{3} \times -\frac{3}{2} = -1\]Since this product equals -1, line segments AB and AQ are perpendicular, confirming a right angle.

Key Concepts

Understanding the Slope of a Line

The slope of a line is a measure of its steepness. It describes how much the line rises or falls as you move from left to right. Here's how to find the slope between two points, say

• \((x_1, y_1)\)
• \((x_2, y_2)\)

You use the slope formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This method tells you how much \(y\) (the vertical change) changes for a change in \(x\) (the horizontal change). If the slope is positive, the line rises. If the slope is negative, the line falls.

In the original exercise, calculating the slopes for each line segment helps us build an understanding of each segment's direction and steepness.

Exploring Perpendicular Slopes

Perpendicular slopes are an interesting geometry concept. They come into play when two lines meet at a right angle. For two lines to be perpendicular, the product of their slopes must be equal to

• -1

In mathematical terms, if the slope of one line is \(m_1\) and the slope of another line is \(m_2\), then:\[m_1 \times m_2 = -1\]This tells us the lines intersect at a right angle, forming what's known as a perpendicular relationship.

In the exercise, checking this condition for the line segments \(AB\) and \(AQ\), we find they are indeed perpendicular as their slopes' product is -1. This is the key to proving the triangle has a right angle, leading us to conclude that it is a right triangle.

Core Geometry Concepts in Right Triangles

Right triangles come up often in geometry. They are defined by one specific characteristic: having exactly one right angle, which measures 90 degrees. Understanding this concept is crucial when analyzing triangles formed by points in a coordinate plane.

In such cases, like in the original exercise with points A, B, and Q, proving two sides are perpendicular confirms the presence of a right angle:

• This simple property sets right triangles apart from other triangles.
• Recognizing right triangles is important as it allows for the use of special geometric principles, such as the Pythagorean theorem, which only applies to right triangles.

By examining slopes and demonstrating perpendicularly, we've effectively shown that our triangle is a right triangle using straightforward geometry concepts.