Problem 59 - Algebra A Combined Function

Question

Solve. Two surveyors need to find the distance across a lake. They place a reference pole at point \(A\) in the diagram. Point \(B\) is 3 meters east and 1 meter north of the reference point \(A .\) Point \(C\) is 19 meters east and 13 meters north of point \(A\). Find the distance across the lake, from \(B\) to \(C\).

Step-by-Step Solution

Answer

The distance across the lake from point B to C is 20 meters.

Step 1: Identify Coordinates of Points

We know the positions of the points relative to point A. The coordinates for point A are (0,0) since it is the reference point. For point B, located 3 meters east and 1 meter north of A, the coordinates are B(3,1). For point C, 19 meters east and 13 meters north of A, the coordinates are C(19,13).

Step 2: Apply the Distance Formula

To find the distance between points B(3,1) and C(19,13), use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where \( (x_1,y_1) \) and \( (x_2,y_2) \) are the coordinates of points B and C.

Step 3: Substitute and Compute

Substitute the coordinates of B and C into the distance formula: \[ d = \sqrt{(19 - 3)^2 + (13 - 1)^2} \] Simplify inside the square root:\[ d = \sqrt{(16)^2 + (12)^2} \]Calculate:\[ d = \sqrt{256 + 144} \] \[ d = \sqrt{400} \] \[ d = 20 \]

Key Concepts

Coordinate Geometry

In coordinate geometry, we describe points on a plane using coordinates, which are essentially address-like numbers for each point. **Coordinates** are given as pairs

• The first number represents the point's position on the x-axis (horizontal direction).
• The second number represents the position on the y-axis (vertical direction).

The origin is the point where the x and y axes intersect, and it is denoted as \((0,0)\). In this exercise, point A is at the origin, making it a reference point.
Using such a system, we methodically plot points like our problem's points B and C, exactly pinpointing their placement relative to A.

Application of Geometry

Geometry serves as a fantastic tool for solving real-world problems like finding the distance across a lake. When applied, it transforms visual problems into calculable solutions. Here, **coordinate geometry** is employed to map out surveyor positions on a lake.

• By creating a diagram, surveyors can accurately initialize points and distances from one another.
• Geometry allows calculation of distances not easily measured directly.

This systematic method enables efficient problem-solving in various settings, ensuring that calculations used for complex measurements are straightforward and accurate.

Distance Calculation

The distance formula in coordinate geometry is a valuable tool for calculating the straight-line distance—or shortest path—between two points. It is derived from the Pythagorean theorem. The formula is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

• Subtract the x-coordinates to find the horizontal distance.
• Subtract the y-coordinates for the vertical distance.
• Square both differences, add them together, and take the square root to obtain the direct distance.

In the example, using this formula helped find the distance from point B to point C across the lake.

Problem-Solving Steps

Effective problem-solving involves several clear steps. To tackle the problem of measuring the distance across a lake, each step ensures accuracy and comprehensibility.

• **Identify:** Start by identifying the coordinates of the points involved—here, points B and C—with respect to the reference point A.


• **Apply:** Use the distance formula, a reliable method simplified from geometry, for calculating the required distance.


• **Substitute and Compute:** Insert the identified coordinates into the formula, compute step-by-step, ensuring each calculation is correct before moving to the next.

This logical sequence not only provides the correct answer but also improves problem-solving skills by reinforcing the methodology behind the calculations.