Problem 94 - Precalculus Essentials

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

The points A, B, and C are collinear, because the sum of the distances from A to B and from B to C equals the distance from A to C.

Step 1: Calculate the distance between points A and B

First calculate the distance between A(1,1+d) and B(3,3+d). Using the distance formula, this distance equals \(\sqrt{(3 - 1)^2 + ((3+d) - (1+d))^2} = \sqrt{4} = 2\).

Step 2: Calculate the distance between points B and C

Next calculate the distance between B(3,3+d) and C(6,6+d). Using the distance formula, this distance equals \(\sqrt{(6 - 3)^2 + ((6+d) - (3+d))^2} = \sqrt{9} = 3\).

Step 3: Calculate the distance between points A and C

Finally calculate the distance between A(1,1+d) and C(6,6+d). Using the distance formula, this distance equals \(\sqrt{(6 - 1)^2 + ((6+d) - (1+d))^2} = \sqrt{25} = 5\).

Step 4: Compare the distances

The sum of the AB and BC distances (2 + 3) equals the AC distance (5). Therefore, A, B, and C are collinear.

Key Concepts

Distance Formula

Understanding the distance formula is crucial for solving a variety of problems in coordinate geometry. It is derived from the Pythagorean theorem, which relates the lengths of the sides in a right-angled triangle.

To calculate the distance between two points, say point A with coordinates \( (x_1, y_1) \) and point B with coordinates \( (x_2, y_2) \), you can use the formula:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
This formula gives you a straight line distance 'as the crow flies', regardless of the grid's orientation. Let's break it down:

• \((x_2 - x_1)\) and \((y_2 - y_1)\) are the horizontal and vertical differences between the points.
• Squaring these differences helps to eliminate any negative values, since distance is always positive.
• We then sum these squared differences, which corresponds to the sum of the squares of the legs of a right triangle.
• Finally, taking the square root finds the actual length of the hypotenuse, which is the direct distance between the two points.

In the textbook exercise, the distance formula is applied to show that points A, B and C are collinear by proving that the sum of distances AB and BC is equal to the distance AC.

Coordinates Geometry

Coordinates geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This method combines algebra and geometry to establish the position of points, lines, and shapes through numerical coordinates.

The fundamental element of coordinate geometry is the use of an ordered pair \( (x, y) \) to denote a point on the plane. Here's how it works:

• The 'x' value represents the position along the horizontal axis, while the 'y' value represents the position along the vertical axis.
• Using these coordinates, we can graph points, lines, and more complex shapes by following the relationship between their algebraic expressions and their geometric representation.
• Important properties like slope, intercepts, and distance can be calculated directly using algebraic equations.

In our problem, the coordinates of points A, B, and C are used within the distance formula to identify whether the points lie in a straight line, which is a common application of coordinate geometry.

Proof in Geometry

Proof in geometry serves as a logical argument that verifies the truth of a conjecture or statement. Using definitions, theorems, and postulates, proofs can be written in several ways, including two-column proofs, paragraph proofs, and flowchart proofs.

In proofs concerning collinearity:

• One method involves showing that the slope between each pair of points is the same. Equal slopes indicate a straight line.
• Another method, as in our exercise, is to show that the sum of distances between consecutive points equals the distance between the first and last points.

Geometric proofs often require a strong foundation of other mathematical concepts, like the distance formula in coordinate geometry. The step by step process in the textbook exercise not only found the distances between points A, B, and C but also cleverly provided proof that they are collinear by showing the distance relationship.