Question

The distance between the points \( (2, y) \) and \( (10, 3) \) is 10 units. Find the value of \( y \).

Hint:

To find the distance between two points, use the distance formula and solve for the unknown.

Answer 1

The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. \] Substitute the given points \( (2, y) \) and \( (10, 3) \) and the distance \( d = 10 \): \[ 10 = \sqrt{(10 - 2)^2 + (3 - y)^2}. \] Simplify: \[ 10 = \sqrt{8^2 + (3 - y)^2} \quad \implies \quad 10 = \sqrt{64 + (3 - y)^2}. \] Square both sides: \[ 100 = 64 + (3 - y)^2. \] Subtract 64 from both sides: \[ 36 = (3 - y)^2. \] Take the square root of both sides: \[ \pm 6 = 3 - y. \] Step 1: Solve for \( y \). 1. \( 6 = 3 - y \) gives \( y = -3 \). 2. \( -6 = 3 - y \) gives \( y = 9 \). Thus, the possible values of \( y \) are \( y = -3 \) and \( y = 9 \).
Conclusion:
The value of \( y \) is either \( -3 \) or \( 9 \).