Determine whether the three points are the vertices of a right...

Question

Determine whether the three points are the vertices of a right triangle. See Example 3.

$(- 2, - 8), (0, - 4), (- 4, - 7)$

Accepted Answer

Hello everyone. We are going to identify if three points were given can form a right triangle. The points were given R 07 negative 42 and negative 96. I'm going to start off by labeling them. So that way as I'm using them I can tell which ones I'm using. So it seems to me the best way to do this is to recall the distance formula which will give us the length between points. The distance formula is the square root of X sub two minus X sub one squared plus Y sub two minus Y sub one squared. So in this problem we're going to use that formula three times because we need to find out the distance between each of these points. So I'm going to label this. I'm going to start with A. B. So finding the length of A. B means I need the distance between their points. So instead of X sub two and X sub one I'm gonna think of X sub A. And X sub B. So the X from the B point is negative four minus the X from the A point which is zero squared plus the Y. From the B. Point. Which is to minus the Y. From the A point which is seven squared. I'm gonna clean that up Inside my parentheses. I have negative four now squared plus negative five squared. When do out the squares. So I have the square root of negative four squared is 16 plus negative five squared which is adding those together. I have the square root of 41 and I don't think it can be simplified but in this problem I'm not going to simplify it, I'm going to just leave it as is because ultimately that'll work out best for us. Next I want to find the distance between B. C. So I'm gonna use the same formula to find the length of B. C. And so I'm gonna say this is X. Sub see so it's gonna be the X. Value from the sea point. So negative nine minus the X. Value from the B point which is negative four squared plus the y. Value from the sea point which is six minus the y. Value from the B point which is two squared. And just like the previous one I'm gonna clean up what's in my parentheses recall that. Double negatives are like adding. So I have the square root. And under that in that first set of parentheses I have negative nine minus a negative four which is the same as saying negative nine plus four which gives me negative five squared Plus my second set of parentheses which is 6 -2 which is four squared. And again I'm going to clean up the squares. So I have the square root of five squared which is plus four squared which is 16. And somehow again I have the square to 41. So I guess we'll wait and see how this plays out. We have one more side to find the length of and that is side A. C. Or if you want to call it C. A. Either one is fine. So this is the again the same formula to find length. So the x value from C is -9 plus the X value from A. Is zero squared. I'm sorry, that was a minus in there. So what should be in my premises is -9 0 squared. And then plus in my second set of parentheses, the X. The Y from the sea point just six minus the Y. From the A point which is seven squared. So I know I misspoken there for a second. So right now we have the equals the square root of underneath parentheses negative nine minus zero squared plus another set of parentheses, that is six minus seven squared. So cleaning up what's in the parentheses again have the square root and then negative nine minus zero is negative nine squared plus six minus seven which is negative one also squared. So now doing the squares under my square root nine squared is plus negative one squared is one. So here I have the square root of 82. Alright, so now that we have all the lengths of our sizes sides, I can stick this into the pythagorean theorem and try to see if that works out recall that when you have the pythagorean theorem which is a squared plus B squared equals c squared C represents our hypotenuse which is our longest side. If it wasn't clear which side is the longest side, I would have to try these in different orders. But in this example it is very clear to me that a C. Would be the longest side as it's the larger number. So, plugging this in, I'm gonna call a B. A. So I have the square root of 41 squared B. C. Which again is the square root of 41 squared. And if this is a right triangle it has to equal AC. Which is the square root of 82 squared mm Recall that a square cancels out a square root. So all of these squares and square roots will cancel each other out. And we're left with 41 plus 41 equals 82. And it does, since it does, we can confidently say that these four or three points do form a right triangle. So answer choice. A. Yes, they do form a right triangle. Have a nice day.

Determine whether the three points are the vertices of a right triangle. See Example 3.
$(- 2, - 8), (0, - 4), (- 4, - 7)$

Key Concepts

Distance Formula

Pythagorean Theorem

Identifying Right Triangles Using Coordinates

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