Question

what is the length of chord $overline{am}$ in $odot o$ below? a. 12 units b. 6 units c. 14.4 units d. 7.2 units

Explanation:

Step1: Recall the property of chords and perpendiculars from the center

A perpendicular from the center of a circle to a chord bisects the chord. Let's consider the right - angled triangle formed by the radius, half - chord, and the perpendicular from the center to the chord.

Step2: Use the Pythagorean theorem

Let the radius of the circle be \(r\), the perpendicular distance from the center to the chord be \(d = 7.2\), and half of the chord length be \(l\). In the right - angled triangle, by the Pythagorean theorem, if we assume the radius \(r\) and we know the perpendicular distance \(d\) from the center to the chord. Since the two perpendicular distances from the center to the chords \(LM\) and \(AN\) are equal and the chords are symmetrically placed with respect to the center, we can consider one of the right - angled triangles. Let's assume the radius is \(r\). We know that for a chord and the perpendicular from the center, if the perpendicular distance from the center to the chord is \(d\) and the radius is \(r\), and half of the chord length is \(l\), then \(l=\sqrt{r^{2}-d^{2}}\). In this case, since the two chords are symmetric and we are not given the radius but we can observe that the two right - angled triangles formed by the perpendiculars from the center to the chords are congruent. The length of the chord \(AN\) is equal to the length of the chord \(LM\). If we consider the right - angled triangle with perpendicular distance \(d = 7.2\) from the center to the chord and assume the radius \(r\). We know that the two chords are symmetric. The length of the chord \(AN\) is \(12\) units.

Answer:

A. 12 units