Question

The distance between the points (4, y) and (12, 3) is 10 units. Find the value of \( y \).

Hint:

Always square both sides carefully in the distance formula; remember it gives two possible values of \( y \).

Answer 1

Step 1: Use the distance formula.
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 2: Substitute given values.
\[ 10 = \sqrt{(12 - 4)^2 + (3 - y)^2} \]
Step 3: Simplify.
\[ 10 = \sqrt{8^2 + (3 - y)^2} \Rightarrow 10^2 = 64 + (3 - y)^2 \] \[ 100 - 64 = (3 - y)^2 \Rightarrow (3 - y)^2 = 36 \]
Step 4: Solve for \( y \).
\[ 3 - y = \pm 6 \] \[ y = 3 - 6 = -3 \quad \text{or} \quad y = 3 + 6 = 9 \] Step 5: Final Answer.
\[ \boxed{y = -3 \text{ or } 9} \]