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

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(8, 2), Q(3, 5)

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 as the ordered pair and three. And then we also have point V as the ordered pair six and nine. Here we have four answer choice options, answer choices A through D each of which state that the distance D is either four units or two multiplied by the square root of 34 units. And the midpoint is either the ordered pair 11 and six or five and three. So to begin this problem, we can start by solving for the distance. And that 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 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 point U and that means that X sub two and Y sub two will take its values from point V. So now just substituting in our given values into our formula, we find that the distance between points U and V is equal to the square root of six minus 16 quantity squared plus nine minus three quantity squared. Now, we just want to start simplifying by first performing subtraction in each set of parentheses. So we have the square root of the quantity of 10 squared plus six squared. And now the next step is to just square both of the values. So we have the square root of 100 plus 36. And now we just want to sum the values. So, so far, we have found that the distance is the square root of 136. And now we want to continue simplifying by factoring out the largest perfect square from 136. So its largest perfect square is four. So we can rewrite our current expression as the square root of four multiplied by 34. Next, we need to recall that the square root of four simplifies to just two. So we can rewrite our expression in its simplified form as two multiplied by the square root of 34. And now we have found that the distance between points U and V is two multiplied by the square root of units. And now that we have found our distance, we can move on to finding the midpoint between our two given points. So we first need to recall the midpoint formula which states 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 we just need to substitute in our given values into the midpoint formula. So we find that midpoint M between the points U and V is equivalent to the ordered pair where the X coordinate is 16 plus six, all divided by two. And the Y coordinate is three plus nine all divided by two. And now we just want to start simplifying by first summing the values in the numerator. So we have the X coordinate as 22 divided by two and the Y coordinate is then 12 divided by two. Now we just want to divide in both coordinates. So our final ordered pair is 11 and six. And with this midpoint, we are left with answer choice B where again the distance is two multiplied by the square root of units. And the midpoint is the ordered pair 11 and six. 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(8, 2), Q(3, 5)

Key Concepts

Distance Formula

Midpoint Formula

Coordinate Geometry Basics

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