Determine whether the three points are the vertices of a right triangle. See Example 3. (-6,-4),(0,-2),(-10,8)

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Hello everyone. We are given three points and asked if they can form a right triangle. The points were given are negative 14. 6 negative eight negative five and 31. And to make sure we can differentiate them. I'm labeling them A. B and C. The first thing I want you to recall is that to find the length of a side. You need to use the distance formula. So this is the distance between points and it is the square root of X. Sub two minus X. Sub one squared plus Y sub two minus Y. Sub one squared. We're gonna use that Formula three times in this problem. First we're gonna find the length of the line that would be formed by A. B. So distance equals or D. Equals the square root. And opening my set of parentheses. And instead of thinking of it as X. Sub two I'm thinking of it as the X. From the B. Point which is going to be negative eight minus the X. From the A point which is negative 14. Close those parentheses squared plus the Y. From the B. Point which is negative five minus the Y. From the A point which is six. Close those parentheses squared now still under the square root symbol. But I'm going to combine what I have in my parentheses. In my first set of parentheses I have negative eight minus negative 14 which is the same as saying negative eight plus 14 Which will give us six squared Plus my second set of parentheses is negative 5 -6 which is negative still squared. Now I'm gonna do out my square. So under my radical I have six squared is plus negative 11 squared. Which is 121. Right now I have the square root of 36 plus Adding those together. I get the square root of 157. Usually I would try to simplify that but I'm gonna leave it like this because I think in our final step this will actually be easier to look at next. I want to find the length of the side that would be formed by points, B and C. And again I'm gonna use the same formula. So length is going to be the distance between the points and that is the square root of the X. Value from C. Which is three minus the X. Value from B. Which is negative eight squared plus the y. Value from C. Which is one minus the Y. Value from B. Which is negative five. Close my parentheses squared. Next keeping my radical symbol, I'm gonna simplify my parentheses. So three minus negative eight is like saying three plus eight which is 11 squared plus my second set of parentheses which is one minus negative five which is just like one plus five which is six squared. So right now I have the square root of 11 squared plus 16 squared. So the square root of 11 squared is plus six squared is 36. This seems familiar When I add those together. I have the square root of 157. Alright, so two sides are the same length. Doesn't tell us they are a right triangle or aren't we need to do the third side. Third side is going to be finding the length of a C. So again we're gonna use the distance formula distance equals the square root. And in my parentheses under the radical I have the X. Value from C. Which is three minus the X. Value from A. Which is negative 14. Close my parentheses squared plus the Y. Value from C. Which is one minus the Y. Value from A. Which is six. Close the parentheses squared So now still under the radical I'm going to simplify my parentheses. First set I have three minus negative 14 Which is like three plus 14. That's 17 squared plus my next set of parentheses, one minus six is negative five. Also squared. Doing out my squares I have the square root of 17 squared is 289 plus negative five squared is 25. So right now under my radical I have 2 89 plus Simplifying this, I have the square root of 289 plus 25 which is 314. So the length of AC. is the square root of 314. Now recall that to check if something is a right triangle. We're going to use the pythagorean theorem and the pythagorean theorem, a squared plus B squared equals c squared. Or that the sum of the squares of the legs are equal to the square of the hypotenuse. Our hypotenuse is always the longest side. So in this case our hypotenuse would be side A. C. So I'm gonna use A B. As A. So that would be the square root Of squared. Side B is going to be B. C, Which is also the square root of 157 squared. And if this is to be a right triangle, that needs to equal The square root of 314 squared. Because we're saying AC. Has to be our hypotenuse or the longest side. So recall that a square and a square root will cancel each other out. So when I take care of the squares and the square roots I'm left with 157 Plus has to equal 314. And it does. And since it does, that means the three points we were given do make a right triangle and it is 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. (-6,-4),(0,-2),(-10,8)

Verified Solution

Key Concepts

Distance Formula

Pythagorean Theorem

Collinearity