Question

1. the distance from the origin to point p is 5 units. give the coordinates of four possible locations for point p.

Explanation:

Step1: Recall the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, the origin is \((0,0)\) and \(d = 5\), so \(5=\sqrt{(x - 0)^2+(y - 0)^2}\), which simplifies to \(x^{2}+y^{2}=25\).

Step2: Find integer solutions

We can find integer values of \(x\) and \(y\) that satisfy \(x^{2}+y^{2}=25\).

• If \(x = 0\), then \(y^{2}=25\), so \(y=\pm5\). The points are \((0,5)\) and \((0, - 5)\).
• If \(y = 0\), then \(x^{2}=25\), so \(x=\pm5\). The points are \((5,0)\) and \((-5,0)\).

Answer:

\((0,5)\), \((0, - 5)\), \((5,0)\), \((-5,0)\)