Question

(part 2 of 2) using the equation from part 1 above, find the value of x.

Explanation:

Step1: Identify the angle relationship

The angle formed by a tangent and a chord is equal to the measure of the inscribed angle on the opposite side of the chord. Also, a straight angle is \(180^\circ\), but here we use the property that the angle between a tangent and a chord is equal to the inscribed angle subtended by the same chord. Wait, actually, the sum of the angle between the tangent and chord (\(6x - 38\)) and the inscribed angle subtended by the chord (but here the arc is \(140^\circ\), so the angle between tangent and chord is equal to half the measure of the intercepted arc? Wait, no, the measure of an angle formed by a tangent and a chord is equal to half the measure of its intercepted arc. Wait, the intercepted arc here: the angle \((6x - 38)^\circ\) and the \(140^\circ\) arc. Wait, actually, the angle between the tangent and the chord is equal to half the measure of the intercepted arc. Wait, no, let's think again. The angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. But here, the angle \((6x - 38)^\circ\) and the \(140^\circ\) arc: wait, maybe the angle between the tangent and the chord is equal to the inscribed angle on the opposite side, but actually, the correct property is that the measure of an angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. Wait, but in this case, the angle \((6x - 38)^\circ\) and the \(140^\circ\) arc: maybe the angle between the tangent and the chord is equal to half the measure of the intercepted arc, but here, the angle \((6x - 38)^\circ\) is equal to half of the arc that is opposite? Wait, no, maybe the angle between the tangent and the chord is equal to the inscribed angle subtended by the same chord. Wait, alternatively, the sum of the angle between the tangent and chord and the angle subtended by the other arc? Wait, no, let's look at the diagram. The angle \((6x - 38)^\circ\) is formed by a tangent and a chord, and the arc inside the circle is \(140^\circ\). The measure of the angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. Wait, the intercepted arc here would be the arc that is not \(140^\circ\). The total circumference is \(360^\circ\), but no, the circle's total arc is \(360^\circ\), but the angle formed by tangent and chord is half the measure of the intercepted arc. Wait, actually, the correct formula is: the measure of an angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. So if the angle is \((6x - 38)^\circ\), and the intercepted arc is \(140^\circ\)? Wait, no, that doesn't make sense. Wait, maybe the angle between the tangent and the chord is equal to the inscribed angle subtended by the same chord, which is half the measure of the intercepted arc. Wait, maybe I made a mistake. Let's think again. The angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. So if the angle is \((6x - 38)^\circ\), then it should be equal to half of the intercepted arc. But in the diagram, the arc inside the angle is \(140^\circ\), so maybe the intercepted arc is \(140^\circ\), so the angle is half of that? No, that would be \(70^\circ\), but that doesn't fit. Wait, maybe the angle between the tangent and the chord is equal to the measure of the inscribed angle on the opposite side, which is half the measure of the intercepted arc. Wait, alternatively, the angle formed by the tangent and the chord is equal to the angle subtended by the chord in the alternate segment. So the angle \((6x - 38)^\circ\) is equal to the inscribed angle subtended by the chord in the alternate segment, which is half the measure of the intercepted arc. Wait, the intercepted arc here is \(140^\circ\), so the inscribed angle would be half of that, which is \(70^\circ\), but that doesn't fit. Wait, maybe the angle \((6x - 38)^\circ\) and the \(140^\circ\) arc: the angle formed by the tangent and the chord is equal to half the measure of the intercepted arc, so if the angle is \((6x - 38)^\circ\), then \(6x - 38 = \frac{1}{2} \times 140\)? No, that would be \(6x - 38 = 70\), then \(6x = 108\), \(x = 18\), but that doesn't seem right. Wait, maybe the angle formed by the tangent and the chord is equal to the measure of the arc that is adjacent? Wait, no, let's check the diagram again. The angle \((6x - 38)^\circ\) is formed by a tangent and a chord, and the arc inside the circle is \(140^\circ\). The other arc would be \(360 - 140 = 220^\circ\), but that's too big. Wait, no, the angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. So the intercepted arc is the arc that is "cut off" by the chord and the tangent. So if the angle is \((6x - 38)^\circ\), then it's equal to half the measure of the intercepted arc. So if the intercepted arc is \(140^\circ\), then the angle would be \(70^\circ\), but that doesn't fit. Wait, maybe the angle \((6x - 38)^\circ\) is equal to the measure of the arc that is opposite, but that doesn't make sense. Wait, alternatively, the angle between the tangent and the chord is equal to the angle subtended by the chord in the alternate segment, which is equal to half the measure of the intercepted arc. So if the intercepted arc is \(140^\circ\), then the angle is \(70^\circ\), but that would mean \(6x - 38 = 70\), so \(6x = 108\), \(x = 18\). But that seems low. Wait, maybe I got the property wrong. Let's recall: the measure of an angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. So if the angle is \((6x - 38)^\circ\), and the intercepted arc is \(140^\circ\), then:

\(6x - 38 = \frac{1}{2} \times 140\)

Wait, no, that would be \(6x - 38 = 70\), so \(6x = 108\), \(x = 18\). But let's check again. Alternatively, maybe the angle formed by the tangent and the chord is equal to the measure of the arc that is adjacent, but that's not the case. Wait, maybe the angle between the tangent and the chord is supplementary to the angle subtended by the other arc? No, that doesn't make sense. Wait, let's look at the diagram again. The angle \((6x - 38)^\circ\) is formed by a tangent and a chord, and the arc inside the circle is \(140^\circ\). The angle formed by the tangent and the chord is equal to half the measure of the intercepted arc. So the intercepted arc is \(140^\circ\), so the angle is \(70^\circ\), but that would mean \(6x - 38 = 70\), so \(6x = 108\), \(x = 18\). But maybe I made a mistake. Wait, alternatively, the angle formed by the tangent and the chord is equal to the measure of the arc that is opposite, but that's not the case. Wait, maybe the angle \((6x - 38)^\circ\) is equal to the measure of the arc that is \(140^\circ\), but that would mean \(6x - 38 = 140\), so \(6x = 178\), \(x = 29.666...\), which is not an integer. So that can't be. Wait, maybe the angle formed by the tangent and the chord is equal to half the measure of the arc that is not \(140^\circ\). The total circle is \(360^\circ\), so the other arc is \(360 - 140 = 220^\circ\), half of that is \(110^\circ\). So \(6x - 38 = 110\), then \(6x = 148\), \(x = 24.666...\), which is also not an integer. Wait, maybe the angle between the tangent and the chord is equal to the angle subtended by the chord in the alternate segment, which is equal to half the measure of the intercepted arc. Wait, maybe the diagram is showing that the angle \((6x - 38)^\circ\) is equal to the arc \(140^\circ\) divided by 2? No, that would be 70. Wait, maybe the angle formed by the tangent and the chord is equal to the measure of the arc that is adjacent, but that's not the property. Wait, I think I made a mistake. Let's recall the correct property: The measure of an angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. So if the angle is \(\theta\), and the intercepted arc is \(A\), then \(\theta = \frac{1}{2}A\). In this diagram, the angle is \((6x - 38)^\circ\), and the intercepted arc is \(140^\circ\), so:

\(6x - 38 = \frac{1}{2} \times 140\)

\(6x - 38 = 70\)

Then, add 38 to both sides:

\(6x = 70 + 38\)

\(6x = 108\)

Divide both sides by 6:

\(x = \frac{108}{6}\)

\(x = 18\)

Wait, but let's check again. If \(x = 18\), then \(6x - 38 = 6*18 - 38 = 108 - 38 = 70\), which is half of 140. So that makes sense. So the angle formed by the tangent and the chord is half the measure of the intercepted arc (140 degrees), so 70 degrees. Therefore, \(6x - 38 = 70\), solving for x gives 18.

Step1: Set up the equation using the tangent-chord angle property

The measure of an angle formed by a tangent and a chord is half the measure of its intercepted arc. Here, the angle \((6x - 38)^\circ\) intercepts the \(140^\circ\) arc, so:
\(6x - 38 = \frac{1}{2} \times 140\)

Step2: Simplify the right-hand side

Calculate \(\frac{1}{2} \times 140\):
\(6x - 38 = 70\)

Step3: Solve for \(x\)

Add 38 to both sides:
\(6x = 70 + 38\)
\(6x = 108\)

Divide both sides by 6:
\(x = \frac{108}{6}\)
\(x = 18\)

Answer:

\(x = 18\)