Determine whether the three points are collinear. See Example 4. ...

Question

Determine whether the three points are collinear. See Example 4. (0,9),(-3,-7),(2,-19)

Accepted Answer

Welcome back. I am so glad you're here. We are given three points and were asked to identify whether or not these three points are co linear. Well, I'd like to start off by labeling these points. I'm going to label the first one point A. So that's 0 12 and also recognizing that zero, that's the X value for point A. So labeling it X sub A and 12, that's the Y value for point A. So I'm going to label that why sub A. Now the next point is going to be point B, that's our negative five, negative two point negative five is the X value for point B. So that's X sub B and negative two is the Y value for point B. So that's Y sub B lastly point C that is four negative 16 for, that's the X value for point C. So that's X sub C and negative 16. That's the Y value for point C. So that's why subsea. Well, now I've got them labeled, how do we figure out if they're co linear? Well, we recall from previous lessons, the distance formula method for identifying whether three points are co linear we're going to figure out the three different line segments. So one will be from point A to point B, another will be from point B to point C. And the last will be from point A to point C. And what we do is we measure the distance for each of those line segments. We identify the two shortest ones. And if the two shortest ones add up to the distance of the longest one, then they are in fact co linear. If they don't add up, then they're not co linear. So we need to recall from previous lessons, the distance formula that is the square root of than we take in parentheses, X sub two minus X sub one and parentheses, square, it add that to new parentheses. Y sub two minus Y sub one, close those parentheses and square. Well, in the distance formula, we've got excess and wise with subs of one and two. But we identified excess and wise with subs of A B and C. Well, I'll show you how that works for the line segment A B. We'll use the A values as our ones and the B values as our twos. So this is equal to the square root of and for X sub two, that's going to be our X sub B which is negative five, the minus R X sub one, which is going to be our X sub A which is zero, close parentheses. Square, it plus new parentheses are Y sub two which is going to be our Y sub B which is negative two minus R Y sub A which is going to be our excuse me, our Y sub one which is our Y sub A which is 12 close parentheses and square. Now it's a matter of simplification, negative five minus zero that equals negative five still squared plus negative two minus 12. That's negative 14 still squared. Now we can do those exponents. Negative five squared is 25 plus negative 14 squared is 196 doing the addition 25 plus 196 that equals 221. So we have the square root of 221 being the length of the line segment A B. Now let's work on B C. This time B will be our first since C will be our seconds. So taking the square root of we have our X sub two, which is going to be our X sub C which is four minus R X sub one, which is going to be our X sub B which is negative five close parentheses and square it add that to our Y sub two which is going to be our Y sub C which is negative 16. Now I'm going to make a little bit more room minus our Y sub one, which is our Y sub B which is negative two, close those parentheses and square it Now simplifying four minus negative five is the same as four plus five still squared plus negative 16 minus negative two is the same as negative 16 plus two still squared. Now doing the addition inside the parentheses, four plus five is nine squared plus negative 16 plus two. That's negative 14 squared. Now doing the squaring nine squared is plus negative 14 squared is still 196. Now doing the addition 81 Plus is going to be 277. So the square root of 277. Alright, now let's work on the length of AC. So A is going to be our first and C is going to be our second. So we've got X sub two which is going to be our X sub C which is four minus our X sub one which is our X sub A which is zero, close parentheses squared plus Y sub two, which is our Y subsea which is negative minus why sub one? Which is why sub A which is 12 close parentheses square. It now simplifying four minus zero that equals four still squared plus negative 16 minus 12. That is negative 28 still squared. Now we can Do the squaring four square to plus negative 28 squared is 784 adding plus 784 that equals 800. That could simplify a little bit, but I'm going to leave it like that for now. So we've got our lengths now. We need to figure out which two are the shortest. Well, the square root of things aren't always totally intuitive. So popping these into a calculator, we can see what they are. Approximately. The square root of 221 is equal to approximately 14.87. The square root of 277 is approximately 16.64 And the square root of 800 is approximately 28.28. So the shortest to our, this squared of 221, about 14.87 and the squared of 277, about 16.64. So we can add those together 14. plus 16.64. Is that equal to the length of the longest one or this to 8.28? And we can see that no, those are not equal to each other. So this is in fact, not three co linear points which is answer choice B while done, we'll catch you on the next one.

Determine whether the three points are collinear. See Example 4. (0,9),(-3,-7),(2,-19)

Key Concepts

Collinearity of Points

Slope Formula

Determinants in Geometry

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