Question

line segment on is perpendicular to line segment ml. what is the length of segment np? 1 unit 2 units 3 units 4 units

Explanation:

Step1: Apply Pythagorean theorem

In right - triangle \(OPL\), we know \(OL = 5\) (radius of the circle) and \(OP = 4\). By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = OL\), \(a=OP\) and \(b = PL\). So \(PL=\sqrt{OL^{2}-OP^{2}}\).

$$PL=\sqrt{5^{2}-4^{2}}=\sqrt{25 - 16}=\sqrt{9}=3$$

Step2: Use the property of perpendicular from the center to a chord

Since \(ON\perp ML\), a line from the center of a circle to a chord bisects the chord. So \(MP = PL\). In right - triangle \(OPN\) (same as \(OPL\) in terms of the relationship with the radius and the segment from the center to the chord), \(NP = PL\) (because of the perpendicular from the center to the chord property).

Answer:

3 units