Find the distance between each pair of points, and give the coord...

Question

Find the distance between each pair of points, and give the coordinates of the midpoint of the line segment joining them. 
P(3, -1), Q(-4, 5)

Accepted Answer

Hey, everyone here we are asked to solve for the length of the line segment which is joined by the two given points. One point is point H with the values 13 and negative five. And the other point is point K which is at negative 17 and 11 and were also asked to identify the coordinates of the midpoint. So here we have four answer choice options. Answer choice A where the length is two times the square root of 13 and the midpoint is at two and negative eight. Answer choice B also with a length of two times the square root of and a midpoint of 15 and eight. Answer choice C has a length of 34 a midpoint at negative two and three. And lastly answer choice D which also has a length of 34 a midpoint at two and negative three. So we want to begin by solving for the length of our line segment which is joined by our two given points. So to find the length, we want to recall the distance formula for two points. So the distance D is equivalent to the square root of the quantity of X sub two minus X up one squared plus the quantity of Y sub two minus Y sub one squared. So now we just want to identify which X and Y value correspond to either of our points. So we can let our first coordinate point, our first given point point H B the X up one and wise up one point. And that means our other given point point K is going to be the X sub two and Y sub two point. So now with our X and Y values identified, we can go ahead and just substitute in their values into our distance formula. So we see that the distance of the points H K is equivalent to the square root of the quantity of X sub two which is negative minus X sub one which is 13. And then we have this quantity squared plus the quantity of Y sub two which is 11 minus Y sub one which is negative five. And again, we, we need to square this quantity. So now we just need to simplify this expression. So we have the square root of and now negative 17 minus 13 gives us the quantity of negative 30 squared. Then we have plus 11 minus negative five is the same as 11 plus five, 11 plus five. And again, we need to still square this quantity continuing to simplify, we have the square root of now negative quantity squared gives us positive 900 plus 11 plus 15 gives 11 plus five, gives us 16 squared. Continuing to simp five by squaring our 16, we now have the square root of 900 plus 256. Adding the terms together, we get the square root of 1156. Now taking the square of this value, we see that the distance of points H K R 34 units. And now the distance between the two points is the same as the length of the line segment because the endpoints are these two given points. So we have found the length to be 34 units. Well, now we also need to identify the coordinates of the midpoint. So we need to recall the formula for a midpoint which is shown as a coordinate, two coordinates. So we have the X coordinates and for the X coordinates of the midpoint, we take X sub one plus X sub two all divided by two. And then for our Y value, we have Y sub one plus Y sub two all divided by. And so now we've already identified our X and Y values so we can go ahead and just plug them into our midpoint formula here. So we have the midpoint is equal to. And now for our X values, we have X sub one which is 13, X Sub two, which is -17. And this is all divided by two and Now, for our Y values, we have Y sub one which is negative five plus Y sub two, which is 11 and this is all divided by two. And now we just need to simplify this midpoint. So 13 plus negative 17 is the same as minus 17 and this is all divided by two for our X point. And then for our Y value, we have negative five plus 11 which gives us six and this is divided by two. Now simplifying further 13 minus 17, gives us negative four, divided by two. And then we again have six divided by two. And now dividing both of our values by two, we see the midpoint of our line is equal to negative two and three. So we have found our length of the line segment to be 34 units and the midpoint to be at negative two and three. So with these two solutions that we found we are left with answer choice. C thanks for watching. I hope you found this video helpful and I'll see you again next time.

Find the distance between each pair of points, and give the coordinates of the midpoint of the line segment joining them. P(3, -1), Q(-4, 5)

Key Concepts

Distance Formula

Midpoint Formula

Coordinate Plane

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