Problem 94 - College Algebra

Question

Show that the points \(A(1,1+d), B(3,3+d),\) and \(C(6,6+d)\) are collinear (lie along a straight line) by showing that the distance from \(A\) to \(B\) plus the distance from \(B\) to \(C\) equals the distance from \(A\) to \(C\).

Step-by-Step Solution

Answer

Yes, the given points A, B and C are collinear. The calculation has shown that AB + BC = AC, which means that these points lie along a straight line.

Step 1: Calculate the Distance from Point A to Point B

Apply the distance formula, which for two points (x1, y1) and (x2, y2), is \(d = \sqrt{(x2-x1)^2 + (y2-y1)^2}\), to find distance from Point A to B. With A(1, 1+d) and B(3, 3+d) we get distance AB = \(\sqrt{(3-1)^2+(3+d-(1+d))^2} = 2\sqrt{2}\)

Step 2: Calculate the Distance from Point B to Point C

Apply the distance formula similarly to find distance from Point B to C. With B(3, 3+d) and C(6, 6+d) we get distance BC = \(\sqrt{(6-3)^2+(6+d-(3+d))^2} = 3\sqrt{2}\)

Step 3: Calculate the Distance from Point A to Point C

Apply once again the distance formula to find distance from Point A to C. With A(1, 1+d) and C(6, 6+d) we get distance AC = \(\sqrt{(6-1)^2+(6+d-(1+d))^2} = 5\sqrt{2}\)

Step 4: Show That AB + BC = AC

Add the distances from A to B and B to C, and show that it is equal to the distance from A to C. Therefore, 2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}

Step 5: Conclusion

Since AB + BC equals AC, we can conclude that the points A, B, and C lie along a straight line, meaning they are collinear as per the definition of collinear points.