Video: Finding the Measure of the Two Arcs Inscribed between Secants Given the Inscribed Angle

Given that, in the shown figure, 𝑦 = (π‘₯ βˆ’ 2) and 𝑧 = (2π‘₯ + 2), determine the value of π‘₯.

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Video Transcript

Given that, in the shown figure, 𝑦 equals π‘₯ minus two and 𝑧 equals two π‘₯ plus two, determine the value of π‘₯.

First, let’s start with what we know. Line segment 𝐷𝐴 and line segment 𝐡𝐴 intersect outside the circle at point 𝐴. Because that is true, we can say the measure of the angle formed by the two lines is one-half the positive difference of the measures of the intercepted arc. In this case, we already know that the measure of the angle formed by these two lines is 50 degrees, but that 50 degrees is equal to 𝑧 degrees minus 𝑦 degrees over two. Then, what we can do is substitute in two π‘₯ plus two in for 𝑧 and π‘₯ minus two in for 𝑦.

Using this equation, we will be able to find the value of π‘₯. If we distribute the negative to the π‘₯ and the negative two, we will have 50 equals two π‘₯ plus two minus π‘₯ plus two divided by two. If we combine like terms, two π‘₯ minus π‘₯ equals positive π‘₯ and two plus two equals four. And so, we can say that 50 equals π‘₯ plus four over two. From there, we can get the two out of the denominator by multiplying both sides by two. We’ll have 100 equals π‘₯ plus four. And if 100 equals π‘₯ plus four and we subtract four from both sides, we see that π‘₯ equals 96.

We know that 𝑦 was equal to π‘₯ minus two. And so, 𝑦 would be 94 degrees. 𝑧 was equal to two π‘₯ plus two. If we multiply 96 by two and then add two, we get 194. 𝑧 was then equal to 194 degrees. If we wanted to check, we can plug these values back in for 𝑧 and 𝑦 in the original equation we wrote, which says 50 degrees equals 𝑧 degrees minus 𝑦 degrees over two. 194 minus 94 divided by two. 194 minus 94 is 100 and 100 divided by two is 50. And so, we can say that, in the given figure, π‘₯ must be equal to 96.

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