Question

in ⊙h, \\(\overarc{ik} \cong \overarc{jk}\\), \\(m\overarc{ik} = (11x + 2)^\circ\\), and \\(m\overarc{jk} = (12x - 7)^\circ\\). what is the measure of \\(\overarc{ikj}\\)? \\(m\overarc{ikj} = \square^\circ\\)

Explanation:

Step1: Set arc measures equal

Since \(\widehat{IK} \cong \widehat{JK}\), their measures are equal. So, \(11x + 2 = 12x - 7\).

Step2: Solve for \(x\)

Subtract \(11x\) from both sides: \(2 = x - 7\). Then add 7 to both sides: \(x = 9\).

Step3: Find measure of \(\widehat{IK}\) and \(\widehat{JK}\)

Substitute \(x = 9\) into \(11x + 2\): \(11(9) + 2 = 99 + 2 = 101^\circ\). So, \(m\widehat{IK} = m\widehat{JK} = 101^\circ\).

Step4: Find measure of \(\widehat{IKJ}\)

The measure of \(\widehat{IKJ}\) is the sum of \(m\widehat{IK}\) and \(m\widehat{JK}\). So, \(m\widehat{IKJ} = 101 + 101 = 202^\circ\)? Wait, no—wait, a circle is \(360^\circ\), but wait, maybe I misread. Wait, no, \(\widehat{IKJ}\) is the arc from I to K to J? Wait, no, actually, in a circle, if \(\widehat{IK}\) and \(\widehat{JK}\) are congruent, and we need to find the measure of \(\widehat{IKJ}\), but maybe I made a mistake. Wait, no—wait, the arcs \(\widehat{IK}\) and \(\widehat{JK}\) are congruent, so their measures are equal. So first, solve for \(x\):

\(11x + 2 = 12x - 7\)

\(2 + 7 = 12x - 11x\)

\(x = 9\)

Then, \(m\widehat{IK} = 11(9) + 2 = 101^\circ\), \(m\widehat{JK} = 12(9) - 7 = 108 - 7 = 101^\circ\) (wait, 12*9 is 108, 108 -7 is 101, correct). Then, the arc \(\widehat{IKJ}\) is the sum of \(\widehat{IK}\) and \(\widehat{JK}\)? Wait, no, that can't be, because a circle is 360, but maybe \(\widehat{IKJ}\) is the major arc? Wait, no, maybe I misinterpret the arc. Wait, the problem says "the measure of \(\widehat{IKJ}\)". Let's check the diagram. The circle has center H, points I, K, J on the circle. So \(\widehat{IK}\) and \(\widehat{JK}\) are congruent. So the arc from I to K to J: so \(\widehat{IK} + \widehat{KJ}\)? Wait, no, \(\widehat{JK}\) is the same as \(\widehat{KJ}\). So \(\widehat{IKJ}\) is \(\widehat{IK} + \widehat{KJ}\) (which is \(\widehat{JK}\)). So \(101 + 101 = 202^\circ\). But wait, a circle is 360, so the minor arc would be 360 - 202 = 158, but that doesn't make sense. Wait, no, maybe I messed up the arc. Wait, no—wait, the problem says "In \(\odot H\), \(\widehat{IK} \cong \widehat{JK}\), \(m\widehat{IK} = (11x + 2)^\circ\), and \(m\widehat{JK} = (12x - 7)^\circ\). What is the measure of \(\widehat{IKJ}\)?"

Wait, maybe \(\widehat{IKJ}\) is the arc from I to K to J, which is the sum of \(\widehat{IK}\) and \(\widehat{JK}\). But let's recalculate:

\(x = 9\), so \(m\widehat{IK} = 11*9 + 2 = 101\), \(m\widehat{JK} = 12*9 -7 = 101\). Then, \(m\widehat{IKJ} = 101 + 101 = 202^\circ\). But that seems large, but maybe it's the major arc. Alternatively, maybe the arc is the minor arc, but that would be 360 - 202 = 158, but that contradicts. Wait, no, maybe I made a mistake in the arc definition. Wait, no—wait, the problem is likely that \(\widehat{IKJ}\) is the arc from I to K to J, which is the sum of \(\widehat{IK}\) and \(\widehat{KJ}\) (which is \(\widehat{JK}\)). So 101 + 101 = 202. But let's check again.

Wait, another approach: the central angles for \(\widehat{IK}\) and \(\widehat{JK}\) are equal. So the central angle for \(\widehat{IK}\) is \(101^\circ\), so the arc \(\widehat{IK}\) is \(101^\circ\), same for \(\widehat{JK}\). Then, the arc \(\widehat{IKJ}\) is the sum of these two arcs, so 101 + 101 = 202. So that's the measure.

Answer:

\(202\)