Question

If the distance between the points $(x, 5)$ and $(2, -3)$ is 17 units, then find the value of $x$.

Hint:

Always use the distance formula $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ to find unknown coordinates when the distance is given.

Answer 1

Step 1: Use the distance formula.
The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Step 2: Substitute the given values.
Here, $(x_1, y_1) = (x, 5)$ and $(x_2, y_2) = (2, -3)$, and $d = 17$. \[ 17 = \sqrt{(2 - x)^2 + (-3 - 5)^2} \] \[ 17 = \sqrt{(2 - x)^2 + 64} \] Step 3: Square both sides.
\[ 289 = (2 - x)^2 + 64 \] \[ 225 = (2 - x)^2 \] Step 4: Solve for $x$.
\[ 2 - x = \pm 15 \Rightarrow \begin{cases} x = -13, & \text{if } 2 - x = 15
x = 17, & \text{if } 2 - x = -15 \end{cases} \] Step 5: Conclusion.
Hence, the values of $x$ are $-13$ and $17$.