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Chapter 36: Chapter 36

Problem 639

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Prove that points \(\mathrm{A}(2,3), \mathrm{B}(4,4)\), and \(\mathrm{C}(8,6)\) are collinear.

Short Answer

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The points A(2,3), B(4,4), and C(8,6) are collinear. This is proven by calculating the slopes of lines AB, AC, and BC. The slope of AB is \(\frac{1}{2}\), calculated using the slope formula \(\frac{y_2 - y_1}{x_2 - x_1}\) (specifically, \(\frac{4-3}{4-2} = \frac{1}{2}\)). The same calculation gives the slope of AC as \(\frac{1}{2}\) (from \(\frac{6-3}{8-2}\)) and the slope of BC as \(\frac{1}{2}\) (from \(\frac{6-4}{8-4}\)). As all three slopes are equal, the points are collinear.
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Calculate the slope of line AB

: We will use the slope formula: \(\frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\). For points A(2,3) and B(4,4), the slope of line AB is: \(\frac{4-3}{4-2} = \frac{1}{2}\)

Step 2: Calculate the slope of line AC

: For points A(2,3) and C(8,6), the slope of line AC is: \(\frac{6-3}{8-2} = \frac{3}{6} = \frac{1}{2}\)

Step 3: Calculate the slope of line BC

: For points B(4,4) and C(8,6), the slope of line BC is: \(\frac{6-4}{8-4} = \frac{2}{4} = \frac{1}{2}\)

Step 4: Compare the slopes

: We have the following slopes: - Slope of line AB = \(\frac{1}{2}\) - Slope of line AC = \(\frac{1}{2}\) - Slope of line BC = \(\frac{1}{2}\) All three slopes have the same value of \(\frac{1}{2}\). Therefore, points A, B, and C are collinear.

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