Problem 66 - Algebra for College Students

Question

Solve each geometric figure problem. Use slope to determine whether the points \((0,-1),(2,5)\) and \((5,4)\) are the vertices of a right triangle.

Step-by-Step Solution

Answer

The points form a right triangle because the slopes of two sides are perpendicular with a product of -1.

Step 1 - Calculate the Slope Between Points (0, -1) and (2, 5)

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For points (0, -1) and (2, 5), let (x_1, y_1) = (0, -1) and (x_2, y_2) = (2, 5). Plugging the values into the formula, \(m_1 = \frac{5 - (-1)}{2 - 0} = \frac{6}{2} = 3\).

Step 2 - Calculate the Slope Between Points (2, 5) and (5, 4)

Using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) for points (2, 5) and (5, 4), let (x_1, y_1) = (2, 5) and (x_2, y_2) = (5, 4). Substituting the values, \(m_2 = \frac{4 - 5}{5 - 2} = \frac{-1}{3} = -\frac{1}{3}\).

Step 3 - Calculate the Slope Between Points (0, -1) and (5, 4)

Again using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) for points (0, -1) and (5, 4), let (x_1, y_1) = (0, -1) and (x_2, y_2) = (5, 4). Substituting the values, \(m_3 = \frac{4 - (-1)}{5 - 0} = \frac{5}{5} = 1\).

Step 4 - Determine Perpendicular Slopes

Two lines are perpendicular if the product of their slopes is -1. Here, slopes are \(m_1 = 3, m_2 = -\frac{1}{3}, m_3 = 1\). Check the products: \(m_1 \cdot m_2 = 3 \cdot (-\frac{1}{3}) = -1\), \(m_2 \cdot m_3 = -\frac{1}{3} \cdot 1 = -\frac{1}{3}\), and \(m_1 \cdot m_3 = 3 \cdot 1 = 3\). Since \(m_1 \cdot m_2 = -1\), the lines are perpendicular.

Step 5 - Conclude the Right Triangle

Since the product of the slopes between points (0, -1) and (2, 5), and points (2, 5) and (5, 4) is -1, these lines are perpendicular. Thus, the given points form a right triangle.

Key Concepts

slope formula

To determine whether points form a geometric figure like a right triangle, we first need to understand the slope formula. The slope of a line measures how steep it is and is calculated by the formula:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, \((x_1, y_1)\)and\((x_2, y_2)\) are any two points on a line.
The slope formula helps us find the steepness of the line segment between these two points.

perpendicular slopes

Understanding perpendicular slopes is vital to determine right triangles. Perpendicular lines have slopes that multiply to \(-1\).
For instance, if the slope of one line is\(m_1\) and the slope of another line is\(m_2\), they are perpendicular if:

• \[ m_1 \cdot m_2 = -1 \]

Let's consider the example where we calculated\(m_1 = 3\) and\(m_2 = -\frac{1}{3}\).When we multiply these slopes:

• \[ 3 \cdot -\frac{1}{3} = -1 \]

This means the lines are perpendicular, showing that the triangle formed by these slopes has a right angle.

geometric figures

Geometric figures include shapes like triangles, rectangles, and circles. Understanding the properties of these shapes is essential. A triangle, for example, can be determined to be a right triangle if one of its angles is exactly 90 degrees.
In coordinate geometry, we often use slopes to check such properties. By showing that two lines are perpendicular, we can confirm the presence of a right angle in a triangle.

coordinate geometry

Coordinate geometry combines algebra and geometry using graphs and coordinates. It allows us to solve geometric problems by plotting points on a graph. Each point is defined by an \((x, y)\) coordinate.
For example, to check if the points \((0,-1)\),\((2,5)\), and\((5,4)\)form a right triangle, we plot them on a coordinate plane and use the slope formula.
This helps in calculating the slopes of the lines that connect these points. If the product of the slopes of two lines is \(-1\), the lines are perpendicular, confirming the geometric relationship.