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(-5, -6), Q(7, -1)

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 midpoint M. Here we are given point U as the ordered pair, negative three and negative 11. And we are also given point V as the ordered pair nine and negative five. Here we have four answer choice options. Answer choice is A through D where the distance is either six multiplied by the square root of five units or five multiplied by the square root of six units. And the midpoint is either the ordered pair three and negative eight or negative three and eight. So to begin this problem, we can start by solving for the distance, which means we first need to recall the formula for the distance between two points, which is given as distance D is equivalent 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 need to determine 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 point U. And that means that X sub two and Y sub two will take its values from point B. And so now we just want to substitute in our given values into the formula. So we have that the distance between points U and V is equivalent to the square root of nine minus negative three quantity squared plus negative five minus negative 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 nine plus three quantity squared plus negative five plus 11 quantity squared. And now we just want to sum the values in each set of parentheses. So we have the square root of squared plus six squared. In now squaring both values, we have the square root of plus 36 in. Now summing the values underneath the square root, we get the square root of 180. And now we want to continue simplifying by factoring the largest perfect square from 180. So we can rewrite our current expression as the square root of 36 multiplied by five. And now we just want to recall that the square root of 36 simplifies to six. So we can rewrite our current expression as its simplified form where we have six multiplied by the square root of five. And so we have found that the distance between points U and V is six multiplied by the square root of five units. And with this, we can move on to the next part of our problem where we need to indicate the coordinates of the midpoint. So we first need to recall the formula for the midpoint, which is that midpoint M is equivalent to 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 just substituting in our giving values, we find that the midpoint between points U and V is given by the ordered pair where the X coordinate becomes negative three plus nine all divided by two. And the Y coordinate is negative 11 plus negative five all divided by two. And now we want to start simplifying by first summing the values in the numerators. So the ordered pair so far has the X coordinate of six divided by two and the Y coordinate becomes negative 16 divided by two. And now dividing in both coordinates, we get the ordered pair three and negative eight. And with this, we have found our midpoint as well as the distance between our two given points. And so we are left with answer choice. A where again the distance is six multiplied by the square root of five units. And the midpoint is the ordered pair three and negative eight. 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(-5, -6), Q(7, -1)

Key Concepts

Distance Formula

Midpoint Formula

Coordinate System

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