Problem 100 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. If the points (10,-3) and (2,-9) lie on the same line, then the point \(\left(-12,-\frac{37}{2}\right)\) also lies on that line.

Step-by-Step Solution

Answer

The statement is true. The point \(\left(-12,-\frac{37}{2}\right)\) does lie on the line determined by the points (10,-3) and (2,-9)

Step 1: Calculate the slope between the two given points

The formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \((y_2 - y_1) / (x_2 - x_1)\). So, the slope between points \((10,-3)\) and \((2,-9)\) is \((-9 + 3) / (2 - 10) = -6 / -8 = 0.75\)

Step 2: Calculate the slope between first given point and the point to be checked

Using the same slope formula, calculate the slope between points \((10,-3)\) and \(\left(-12,-\frac{37}{2}\right)\). Substituting into the formula it will be \(\left(-\frac{37}{2} + 3\right) / (-12 - 10) = -\frac{31}{2} / -22 = 0.75\)

Step 3: Compare the slopes

The slopes calculated in step 1 and step 2 are the same, 0.75. Because the slopes of the line segments from the first point to the second point, and from the first point to the third point are the same, we can conclude that the three points lie on the same line

Key Concepts

Slope of a Line

Understanding the slope of a line is essential in coordinate geometry. It tells us how steep the line is, and the direction it goes.The slope is often represented by the letter \(m\) and is calculated by taking the difference in the \(y\)-coordinates, divided by the difference in the \(x\)-coordinates, between two points. The formula to find the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]If both the numerator and the denominator are negative, the slope becomes positive, as dividing two negative numbers results in a positive fraction. This concept was used to calculate the slope between the points (10,-3) and (2,-9), resulting in 0.75. A slope of 0.75 means that for every one unit you move to the right horizontally, you move 0.75 units up vertically. Understanding how to compute the slope gives you a powerful tool for analyzing the relationships between different points on a graph.

Collinear Points

Collinear points are points that lie on the same straight line. One way to determine if three points are collinear is to compare the slopes between pairs of points.If the slopes between every pair of points are equal, then the points are collinear.In this exercise, the points (10,-3), (2,-9), and \((-12,-\frac{37}{2})\) were examined. Both slopes, calculated from different pairs of these points, were found to be 0.75. Since the slopes are equal, we conclude these points are indeed collinear.When calculating with precise values or fractions, ensure the calculations for slopes are accurate. This is crucial because even a small mistake can alter the determination of collinearity.Collinear points are significant in geometry as they simplify understanding the geometric alignment and relationships between points.

Equation of a Line

The equation of a line is a mathematical way to describe a line in a coordinate plane. It usually takes the form \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept.The y-intercept is the point where the line crosses the y-axis.To find the equation, you need a slope and a point on the line. For example, using the point (10, -3) and the slope 0.75:1. Use the point-slope formula: \[y - y_1 = m(x - x_1)\] Substitute the point and slope: \[y + 3 = 0.75(x - 10)\]2. Simplify to get the line equation in standard form (expand and solve for \(y\)): \[y = 0.75x - 0.75 \times 10 - 3\] \[y = 0.75x - 10.5\]This equation tells you that any value of \(x\) you choose will give you the corresponding \(y\), which lies on this line. This concept is crucial for predicting and understanding the characteristics of the line across the coordinate plane.