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Chapter 2: Problem 119

Geometry Use slopes to show that the triangle whose vertices are \((-2,5),(1,3),\) and (-1,0) is a right triangle.

Short Answer

Expert verified
The triangle is a right triangle because sides AB and BC are perpendicular.

Step by step solution

01

- Find the slopes of the sides

To determine if a triangle is a right triangle using slopes, begin by identifying the coordinates of the vertices. The vertices are given as \((-2, 5)\), \(1, 3\), and \((-1, 0)\). Next, calculate the slopes of each of the three sides using the formula for the slope between two points, \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
02

- Calculate the slope of side (1, 3) to (-2, 5)

Using the formula for slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\), we find: \[ m_{AB} = \frac{3 - 5}{1 - (-2)} = \frac{3 - 5}{1 + 2} = \frac{-2}{3} \] Therefore, the slope of side AB is \(-\frac{2}{3}\).
03

- Calculate the slope of side (1, 3) to (-1, 0)

Using the formula for slope: \[ m_{BC} = \frac{3 - 0}{1 - (-1)} = \frac{3 - 0}{1 + 1} = \frac{3}{2} \] Therefore, the slope of side BC is \(\frac{3}{2}\).
04

- Calculate the slope of side (-2, 5) to (-1, 0)

Using the formula for slope: \[ m_{CA} = \frac{0 - 5}{-1 - (-2)} = \frac{0 - 5}{-1 + 2} = \frac{-5}{1} = -5 \] Therefore, the slope of side CA is \-5\.
05

- Determine if the slopes are perpendicular

For a triangle to have a right angle, the slopes of two sides must be negative reciprocals of each other (meaning their product must be \(-1\)). Check the products of the slopes we have calculated:1. \(-\frac{2}{3} \times \frac{3}{2} = -1\)2. \(-\frac{2}{3} \times (-5) = \frac{10}{3}\) (not equal to \(-1\))3. \(-5 \times \frac{3}{2} = -\frac{15}{2}\) (not equal to \(-1\))The slopes \(-\frac{2}{3}\) and \(\frac{3}{2}\) are negative reciprocals, indicating that sides AB and BC are perpendicular, forming a right angle at \(B(1, 3)\).

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Key Concepts

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

slope formula
To understand if a triangle is a right triangle using geometry, you start with the slope formula. The slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. This formula helps us measure the steepness or inclination of a line connecting the two points.

For example, consider finding the slope between \( (1, 3) \) and \( (-2, 5) \). Using the formula, we plug in the values:
\[ m = \frac{3 - 5}{1 - (-2)} = \frac{-2}{3} \]. This tells us that the slope is \(-2/3\).

Calculating slopes for each side of the triangle is essential in determining the type of triangle we are dealing with.
perpendicular slopes
When two lines are perpendicular, their slopes have a specific relationship: they are negative reciprocals of each other. This means that if one line has a slope of \( m_1 \), the perpendicular line will have a slope of \(-1/m_1\), and their product will be \(-1\).

For example, let's verify this with the slopes calculated from the vertices of our triangle: \(-2/3\), \(3/2\), and \(-5\).

Multiplying the slopes \( -2/3 \) and \ (3/2) \:
\[ \left( -\frac{2}{3} \right) \cdot \left( \frac{3}{2} \right) = -1 \].

The product is indeed \(-1\), which confirms that these sides are perpendicular. In our context, this means side AB and BC are perpendicular, forming a right angle at B.
right triangle proof
The next step is to prove that a triangle is a right triangle using our findings. We already calculated the slopes of the sides and found that \(-2/3\) and \ (3/2) \ are negative reciprocals, which means they form a right angle. Since our definition of a right triangle includes having one 90-degree angle, this verification is crucial.

Therefore, with \ (1, 3) \ as the common vertex, sides AB and BC form a right angle.

Conclusively, the triangle with vertices \ (-2, 5), (1,3), and (-1,0) \ is a right triangle. Summarized steps:
  • Calculate the slope of each side using the slope formula
  • Check the product of slopes to identify perpendicular sides
  • Verify the presence of a 90-degree angle

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