Question

Find the value of $x$ for which the distance between the points $(x,2)$ and $(6,5)$ is 5 units.

Hint:

Always apply distance formula $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Squaring both sides helps eliminate the square root, but remember to check both positive and negative solutions.

Answer 1

By distance formula: \[ \sqrt{(x-6)^2+(2-5)^2}=5 \] \[ \sqrt{(x-6)^2+(-3)^2}=5 \] \[ \sqrt{(x-6)^2+9}=5 \] Square both sides: \[ (x-6)^2+9=25 \] \[ (x-6)^2=16 \] \[ x-6=\pm4 \] \[ x=6+4=10 \text{or} x=6-4=2 \] \[ \boxed{x=10 \ \text{or} \ x=2} \]