Question

$\overleftrightarrow{hj}$ is tangent to $\odot g$. what is $hj$?
(image of a circle with center g, tangent segment hj, radius gh = 16 mi, and segment gj = 20 mi)
$hj = \square$ mi

Explanation:

Step1: Use the Pythagorean theorem

Since \( \overleftrightarrow{HJ} \) is tangent to \( \odot G \), \( \angle GHJ = 90^{\circ} \) (tangent - radius property). In right - triangle \( GHJ \), by the Pythagorean theorem \( a^{2}+b^{2}=c^{2} \), where \( c = GJ = 20\) mi (hypotenuse), \( a = GH=16\) mi (one leg), and \( b = HJ\) (the other leg).
So, \( HJ^{2}+GH^{2}=GJ^{2} \), which can be rewritten as \( HJ=\sqrt{GJ^{2}-GH^{2}} \).

Step2: Substitute the values

Substitute \( GJ = 20\) and \( GH = 16\) into the formula:
\( HJ=\sqrt{20^{2}-16^{2}}=\sqrt{(20 + 16)(20 - 16)}\) (using \(a^{2}-b^{2}=(a + b)(a - b)\))
\(=\sqrt{36\times4}=\sqrt{144}\)
\( = 12\)

Answer:

\(12\)