Question

using the diagram, find the length of \\( \overline { m n } \\) if we know \\( \overline { l o } \\) has a length of 1.5 in. and \\( \overline { m o } \\) has a length of 8.2 in.

Explanation:

Step1: Recall the property of tangent to a circle

A tangent to a circle is perpendicular to the radius at the point of tangency. So, \(\angle OLM = \angle ONM=90^{\circ}\).

Step2: Use the Hypotenuse - Leg (HL) congruence criterion for right - angled triangles

In right - angled triangles \(\triangle OLM\) and \(\triangle ONM\), \(OL = ON\) (radii of the same circle), \(OM = OM\) (common side). By HL congruence, \(\triangle OLM\cong\triangle ONM\).

Step3: Apply the Pythagorean theorem

In right - angled triangle \(\triangle OLM\), by the Pythagorean theorem \(LM=\sqrt{OM^{2}-OL^{2}}\). Given \(OL = 1.5\) in and \(OM = 8.2\) in. Then \(LM=\sqrt{8.2^{2}-1.5^{2}}=\sqrt{(8.2 + 1.5)(8.2-1.5)}=\sqrt{9.7\times6.7}=\sqrt{64.99}\approx 8.1\) in.
Since \(\triangle OLM\cong\triangle ONM\), \(MN = LM\).

Answer:

The length of \(\overline{MN}\) is approximately \(8.1\) in.