What is the value of ( r ) if the Distance Between the Points is 3 Units?

When dealing with problems involving distance between points in coordinate geometry, the distance formula is a powerful tool. This article will guide you through the process of finding the value of ( r ) in a specific problem where the distance between the points (4, 2) and (1, ( r )) is 3 units. Understanding the distance formula and how to solve the resulting equation will be crucial.

Understanding the Problem

The problem can be summarized as follows: Given the points (4, 2) and (1, ( r )), find the value of ( r ) such that the distance between these points is 3 units. This problem requires the application of the distance formula, which is given by:

Distance Formula: ( d sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2} )

Step-by-Step Solution

Let's start by assigning the coordinates of the given points to the variables in the distance formula:

The coordinates of the first point are (4, 2), so ( x_1 4 ) and ( y_1 2 ). The coordinates of the second point are (1, ( r )), so ( x_2 1 ) and ( y_2 r ).

The distance between these points is given as 3 units. Plugging these values into the distance formula:

[ d sqrt{(1 - 4)^2 (r - 2)^2} ]

Since the distance ( d ) is 3, we can write:

[ 3 sqrt{(1 - 4)^2 (r - 2)^2} ]

To simplify the equation, we first simplify inside the square root:

[ 3 sqrt{(-3)^2 (r - 2)^2} ]

[ 3 sqrt{9 (r - 2)^2} ]

Solving the Equation

To eliminate the square root, we square both sides of the equation:

[ 3^2 9 (r - 2)^2 ]

[ 9 9 (r - 2)^2 ]

Subtract 9 from both sides:

[ 0 (r - 2)^2 ]

Since the square of a number is zero only if the number itself is zero, we have:

[ r - 2 0 ]

Solving for ( r ), we get:

[ r 2 ]

Conclusion

Thus, the value of ( r ) is 2. To verify this, we can check if the distance between the points (4, 2) and (1, 2) is indeed 3 units:

[ d sqrt{(1 - 4)^2 (2 - 2)^2} ]

[ d sqrt{(-3)^2 0^2} ]

[ d sqrt{9 0} ]

[ d sqrt{9} ]

[ d 3 ]

This confirms our solution. Therefore, the value of ( r ) is indeed 2.

Additional Notes

It's worth noting that the problem can be approached geometrically as well. By plotting the point (4, 2) and drawing a circle of radius 3, you can read off the y-coordinate of the point (1, ( r )) on this circle, which is also 2. This confirms our algebraic solution.

Key Takeaways

The distance formula is a geometric representation of the Pythagorean theorem. It is essential to isolate the variable in the equation before solving it. Checking the solution geometrically can provide further validation and understanding.

Understanding and applying the distance formula in this manner can be a valuable skill in coordinate geometry and problem-solving in mathematics.