For the points P and Q, find ( a ) the distance d(P, Q) and ( b ) the coordinates of the midpoin

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

Key Concepts

Distance Formula

The distance formula is derived from the Pythagorean theorem and is used to compute the distance between two points in a coordinate plane. Given any two points with coordinates (x?, y?) and (x?, y?), the distance between them is calculated using the formula ?((x? - x?)² + (y? - y?)²), which provides a measure of the straight line distance connecting the points.

Midpoint Formula

The midpoint formula is used to determine the exact center point of the line segment connecting two points in a coordinate plane. For points (x?, y?) and (x?, y?), the midpoint’s coordinates are given by ((x? + x?)/2, (y? + y?)/2), which gives the average of the x-coordinates and the y-coordinates, effectively dividing the segment into two equal halves.

Transcript

00:01 In this problem, we are given points p and q and then ask to find the distance and midpoint between those two points.

00:09 So first of all, let's plot points p and q.

00:14 Point p is negative square root of 7, comma, 8 times a square root of 3.

00:21 Q is 5 times the square root of 7 times a negative square root of 3.

00:25 Putting these into a more digestible format, negative square root of 3.

00:31 7 is about negative 2 .6.

00:36 This 13 .86, q, 5 times the square root of 7 is 13 .23, comma, negative 1 .73.

00:51 Okay, so now that we can, you know, easily see these, let's plot these points.

00:56 So negative 2 .6, that's 2 .6 to the left of the y -axis.

01:02 Then up 13 .86.

01:05 That's going to be off the graph up here.

01:08 Let me make this, get myself some room here.

01:14 But we don't really need those numbers now anyway.

01:18 We have point p plotted.

01:20 And now for point q, it's 13 .23.

01:23 Again, 13 .23 is somewhere around over here.

01:28 And negative 1 .7.

01:29 That's one about right there.

01:33 So here we've plotted all of the points and let me connect these and label it.

01:43 So that's p, that's q.

01:47 And just before we get started here, you can see that the midpoint of this line is about right here.

01:57 So once we figure out the number, using the formula in the top right, we can actually you know verify that it is about right there so moving on to part a is finding the distance between points p and q and we'll use this formula up here it if you have a good eye you can see that it looks an awful lot like the pythagorean theorem that's c squared oh c squared let me try this somewhere else.

02:42 C squared equals a squared plus b squared and then to find c which is our distance between points p and q we can just take the square root of both sides.

02:55 So that's just a little quick basic explanation of what we're doing here.

03:03 So here the distance between points p and q is equal to the square root of x2 minus x1 so these subscripts correspond to the points with q being subscript two and p being one so we take the x value of two which is q so five times a square root of 7 and then subtract a negative square root of 7 and then quantity squared.

03:45 There's your, you know, a squared or yeah, a squared.

03:51 And then the y values will do negative square root of 3 plus 8 times a square root of 3, quantity squared, all under that root...

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