Question:

Find the value of \( k \) if the distance between \((k, 3)\) and \((2, 3)\) is 5.

Choices:

5

6

7

8

Correct Answer: 7

Explanation:

To determine the value of \( k \), we use the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a Cartesian plane:

\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Here, the given points are \((k, 3)\) and \((2, 3)\) and the distance \( D \) between them is 5. The distance formula can be simplified, considering that both points have the same \( y \)-coordinate (3):

\[ 5 = \sqrt{(2 - k)^2 + (3 - 3)^2} \]

This simplifies to:

\[ 5 = \sqrt{(2 - k)^2 + 0} \]

Thus:

\[ 5 = \sqrt{(2 - k)^2} \]

We can eliminate the square root by squaring both sides:

\[ 5^2 = (2 - k)^2 \]

Which simplifies to:

\[ 25 = (2 - k)^2 \]

To solve for \( k \), we take the square root of both sides again:

\[ \sqrt{25} = |\sqrt{(2 - k)^2}| \]

So:

\[ 5 = |2 - k| \]

This implies two potential equations for the absolute value:

\[ 2 - k = 5 \quad \text{or} \quad 2 - k = -5 \]

Solving the first equation:

\[ 2 - k = 5 \implies -k = 5 - 2 \implies -k = 3 \implies k = -3 \]

Solving the second equation:

\[ 2 - k = -5 \implies -k = -5 - 2 \implies -k = -7 \implies k = 7 \]

The feasible and correct value for \( k \) from the given options is 7.

Thus, the correct answer is 7.