Question

select the point(s) that satisfy the condition. two points on the x-axis that are 10 units from (1, 8) a) (5, 0), (-7, 0) b) (-5, 0), (-7, 0) c) (5, 0), (7, 0) d) (-5, 0), (7, 0)

Explanation:

Step1: Recall Distance Formula

The distance \(d\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For points on the \(x\)-axis, \(y = 0\), so let the point be \((x,0)\). The given point is \((1,8)\), and distance \(d = 10\).

Step2: Substitute into Formula

Substitute \((x_1,y_1)=(1,8)\), \((x_2,y_2)=(x,0)\), and \(d = 10\) into the distance formula:
\(10=\sqrt{(x - 1)^2+(0 - 8)^2}\)
Square both sides: \(100=(x - 1)^2 + 64\)
Subtract 64: \((x - 1)^2=36\)
Take square roots: \(x - 1=\pm6\)

Step3: Solve for \(x\)

Case 1: \(x - 1 = 6\Rightarrow x = 7\)
Case 2: \(x - 1=-6\Rightarrow x=-5\)
So the points are \((-5,0)\) and \((7,0)\), which is option D.

Answer:

D. \((-5, 0)\), \((7, 0)\)