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Chapter 1: Problem 54

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

Short Answer

Expert verified
AB and CA are perpendicular, confirming a right triangle.

Step by step solution

01

Calculate Slope of AB

To find the slope of line segment AB, we need the coordinates of points A and B. The slope formula 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} \]. \Given the coordinates of A(-3, -1) and B(3, 3), the slope of AB is \\[ m_{AB} = \frac{3 - (-1)}{3 - (-3)} = \frac{4}{6} = \frac{2}{3} \].
02

Calculate Slope of BC

Next, we'll find the slope of the line segment BC using the formula for slope. \The coordinates for points B(3, 3) and C(-9, 8) are used here. \\[ m_{BC} = \frac{8 - 3}{-9 - 3} = \frac{5}{-12} = -\frac{5}{12} \].
03

Calculate Slope of CA

Now let's calculate the slope of the line segment CA. Using the points C(-9, 8) and A(-3, -1), the slope is calculated as \\[ m_{CA} = \frac{-1 - 8}{-3 - (-9)} = \frac{-9}{6} = -\frac{3}{2} \].
04

Check for Perpendicular Slopes

In a right triangle, the slopes of two sides must be negative reciprocals if they are perpendicular. We'll check between the pairs of segments: \- The negative reciprocal of \( \frac{2}{3} \) is \( -\frac{3}{2} \). \- The slope \( m_{CA} = -\frac{3}{2} \) matches this reciprocal. \Thus, segments AB and CA are perpendicular.

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Key Concepts

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

Slope Calculation
The slope of a line represents its steepness and is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula uses two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line:
  • The numerator \( y_2 - y_1 \) represents the change in the y-values.
  • The denominator \( x_2 - x_1 \) is the change in the x-values.
  • The fraction \( \frac{\text{change in } y}{\text{change in } x} \) gives the slope.
The slope is important because it tells us how the y-value changes for every unit change in x. If the slope is positive, the line goes upwards as you move from left to right. If it is negative, the line goes downwards. Understanding slope calculations can help in determining relationships between lines, such as parallelism or perpendicularity, key to identifying right triangles when these slopes reveal perpendicular peaks.
Perpendicular Lines
Perpendicular lines intersect at a right angle (90 degrees) and a fundamental way to identify them is by examining their slopes. For two lines to be perpendicular, the product of their slopes must equal -1.
  • If one line has a slope \( m_1 \), the perpendicular line will have a slope \( m_2 \) such that \( m_1 \times m_2 = -1 \).
  • If \( m_1 = \frac{2}{3} \), then the perpendicular slope \( m_2 \) must be \( -\frac{3}{2} \).
This negative reciprocal relationship is convenient for verifying right triangles in coordinate geometry, as one can calculate slopes to check if exactly two sides are perpendicular, confirming a 90-degree angle with precision. Remember, perpendicularity is a characteristic of right triangles, and recognizing this geometrical feature is invaluable.
Coordinate Geometry
Coordinate geometry uses algebraic techniques to study geometric relationships. This method involves plotting points, lines, and shapes on the Cartesian plane.
  • Points are defined by their coordinates \((x, y)\).
  • Lines and slopes connect these points mathematically.
  • Geometric properties like distances, midpoints, and angles can be calculated.
In the context of verifying right triangles, coordinate geometry enables precise analysis by using coordinates to ascertain sides' slopes and determine their perpendicularity. It provides a bridge between algebra and geometry, offering a powerful toolset to evaluate geometrical structures analytically. This approach is especially useful for students, as it helps them transition from visual recognition of figures to analytical proof using numerical and algebraic data.

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