For the points P and Q, find (a) the distance d(P, Q) and (b) the...

Question

For the points P and Q, find (a) the distance d(P, Q) and (b) the coordinates of the mid-point M of line segment PQ. See Examples 2 and 5(a). P(-5,-6), Q(7,-1)

Accepted Answer

Hello everyone. We are going to find the length and midpoint of the line segment that would join the points A. Is negative four negative eight and B. Which is six negative two. The length of this is also sometimes referred to as the distance. And you may recall the distance formula for between points is the square root of X. Sub two minus X. Sub one squared plus Y sub two minus Y sub one squared. I'm gonna take and label my A point as X. Sub one. Y sub one. M. I. B. Point is X. Up to Y sub to recall that every time you have a point you have an X. Value and a Y. Value. So going back to my distance formula which we will use to find the length. I'm going to plug in the values from my points. So it's the square root of X. Sub two which is six minus X sub one which is negative four squared plus Y sub two which is negative two minus Y sub one which is negative eight squared. No my double negatives. This is because I'm subtracting a negative number. I can't just pretend that one of them wasn't negative because it already had a minus sign both belong there and we're going to end up using them. So next step I recall that my double negatives are like adding and that's what I'm going to do to clean up inside the parentheses. So six minus and negative four is the same as saying six plus four Which is 10 squared. And then plus my next set of parentheses. Negative two minus negative eight is the same as saying negative two plus eight which would give us a positive six also squared. Now I'm going to do those squares. So still under the square root symbol 10 squared is 100 plus six squared which is 36. And then adding them together, I have the square root of and I want to break that down to the best of my ability so I'm gonna find a perfect square that I can factor out so I can simplify this radical. I don't know all of the perfect squares but I know that 1 36 isn't one. So I'm thinking of a perfect square that could go evenly into it. After checking a couple, I've realized that four times 34 multiplies to 136. So the square to four times the square to 34 will help me simplify this radical. The square root of four is 2. And you cannot simplify the Square 2 30 for any further. So our distance or length is too radical. 34 units. So that's the first part of our solution. Not done yet because we still need to find the midpoint recall that the midpoint formula is again an X and a Y. So we're gonna end up with a point and it is formed by doing X. Sub one plus X. Sub two divided by two for the X value. And why? Sub one plus Y. Sub two divided by two for the Y value. So plugging in our values, X sub one is negative four Plus X. Sub two which is six divided by two. And then we have Y sub one which is negative eight plus Y sub two which is negative two divided by two. Negative four plus six is a positive two. So we have 2/2. Negative eight plus negative two is negative 10 divided by two. So our midpoint looks like two divided by two is 1 negative 10 divided by two is negative five. So our mid point is the 50.0.1 negative five. So now that we have both parts of our solution, let's check to see which answer choice that is and it appears to be answer choice C Where we have the length of two times radical 34 units and the midpoint of -5. Have a nice day.

For the points P and Q, find (a) the distance d(P, Q) and (b) the coordinates of the mid-point M of line segment PQ. See Examples 2 and 5(a). P(-5,-6), Q(7,-1)

Key Concepts

Distance Formula

Midpoint Formula

Coordinate Plane

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