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 midpoint M of line segment PQ. See Examples 1 and 2.

P(-6, -5), Q(6, 10)

Accepted Answer

Hey, everyone here we are asked to solve for the distance D between the two points given below and indicate the coordinates of the midpoint M. Here we are given point U which is the ordered pair negative three and negative 11. Then we are also given point V which is the ordered pair five and nine. Here we have four answer choice options, answer choices A through D each of which state that the distance is either 29 multiplied by the square root of two units or four multiplied by the square root of 29 units. And the midpoint is either the ordered pair four and negative 10 or one and negative one. So we can begin this problem by first solving for the distance between our two given points. And so we first need to recall the formula for the distance between two points which is given as the distance D is equal to the square root of X sub two minus X, sub one quantity squared plus Y, sub two minus Y sub one quantity squared. And now we just want to identify our values for X and Y in relation to our given points. So we can let X sub one and Y sub one take its values from our first given point which is point U. And that means that X sub two and Y sub two will get its values from point B. So now just substituting in our given values into our formula, we have that the distance between U and V is equivalent to the square root of five minus negative three quantities squared plus nine minus negative 11 quantity squared. And now we just want to start simplifying by first simplifying the expressions within each set of parentheses. So we have the square root of five plus three quantities squared plus nine plus 11 quantity squared. And now summing our values, we have the square root of eight squared plus 20 squared. And now squaring both values, we have the square root of 64 plus 400. And now the next step is to just sum our remaining values. So we have the square root of 464. And now we want to continue simplifying our expression further by factoring out the largest perfect square from the value underneath the square root. So the largest perfect square of 464 is 16. So we can rewrite our current expression as the square root of 16 multiplied by 29. And now we just need to recall that the square root of simplifies to four. So we can now simplify our current expression by rewriting it as four multiplied by the square root of 29. So now we have found that the distance between our two points is for multiplied by the square root of 29 units. And so now that we have identified our distance, we can move on to indicating the coordinates of our midpoint. So we first need to recall the midpoint formula which states that midpoint M is given by the ordered pair where the X coordinate is X sub one plus X sub two all divided by two. And the Y coordinate is Y sub one plus Y sub two all divided by two. And now we just want to substitute in our given values into the midpoint formula. So we have that the midpoint from the points U and V is given by the ordered pair where the X coordinate is negative three plus five all divided by two. And the Y coordinate is negative 11 plus nine all divided by two. And now we need to start simplifying by first summing the values in both of the numerators. So we have the ordered pair where the X coordinate is now two divided by two and the Y coordinate is negative two divided by two. And now dividing in both coordinates, we have the final ordered pair one and negative one. So now we have identified the midpoint of our two given points. So we are left here with answer choice C where again, the distance is four multiplied by the square root of 29 units. And the midpoint is the ordered pair one and negative one. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

For the points P and Q, find (a) the distance d(P, Q) and (b) the coordinates of the midpoint M of line segment PQ. See Examples 1 and 2. P(-6, -5), Q(6, 10)

Key Concepts

Distance Formula

Midpoint Formula

Coordinate System

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