Question

line segment on is perpendicular to line segment ml. what is the length of chord ml? line segment om = 13 units in length, line segment pn = 8 units in length. 20 units 24 units 26 units 30 units

Explanation:

Step1: Use the property of perpendicular from the center to a chord

If a line from the center of a circle is perpendicular to a chord, it bisects the chord. So \(ML = 2\times ML_{half}\) (where \(ML_{half}\) is the length of either \(MP\) or \(PL\)). Also, \(ON = OM\) (radii of the same circle), so \(ON=13\) units. Given \(PN = 8\) units.

Step2: Apply the Pythagorean theorem in right - triangle \(OPM\)

In right - triangle \(OPM\), by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = OM = 13\) (hypotenuse) and \(a=OP\) (we know \(ON = 13\) and \(PN = 8\), so \(OP=ON - PN=13 - 8 = 5\) units), and \(b = MP\).

$$MP=\sqrt{OM^{2}-OP^{2}}$$

$$MP=\sqrt{13^{2}-5^{2}}=\sqrt{(13 + 5)(13 - 5)}=\sqrt{18\times8}=\sqrt{144}=12$$

Step3: Calculate the length of chord \(ML\)

Since \(ML = 2\times MP\) (because the perpendicular from the center to the chord bisects the chord)

$$ML=2\times12 = 24$$

Answer:

24 units