Plotting, Distance, and Midpoint, (a) plot the points, (b) find the distance between the points,

step 1 We are given two points. These points are negative 16 .8, 12 .3, and 5 .6, 4 .9. In Part A, we a

Plotting, Distance, and Midpoint, (a) plot the points, (b)...

Key Concepts

Distance Formula

The distance formula is a method used to determine the length of a line segment between two points in the coordinate plane. Derived from the Pythagorean theorem, it calculates the distance by finding the square root of the sum of the squared differences between the corresponding x-coordinates and y-coordinates.

Midpoint Formula

The midpoint formula is used to find the exact center point of a line segment connecting two points in the coordinate plane. It works by averaging the x-coordinates and the y-coordinates of the two endpoints, providing a simple method to locate the middle of the segment.

Coordinate Plane

The coordinate plane is a two-dimensional surface defined by a horizontal axis (the x-axis) and a vertical axis (the y-axis). It is used to graphically represent points, lines, and curves, enabling visualization of algebraic relationships in a spatial context.

Plotting Points

Plotting points involves placing a dot on the coordinate plane based on specified x (horizontal) and y (vertical) coordinates. This foundational skill allows for the graphical representation of data and helps in visualizing geometric and algebraic relationships.

Transcript

00:02 We are given two points.

00:05 These points are negative 16 .8, 12 .3, and 5 .6, 4 .9.

00:15 In part a, we are asked to plot these points.

00:20 Notice that the x coordinates of the points range from negative 16 .8 up to 5 .6, and the y coordinates from 4 .9 to 12 .3.

00:32 So i'm going to draw the x coordinate, the x -axis, ranging from a nice whole number, let's say negative 20 to positive 10.

01:02 And our y -axis, i'll simply only draw from 0 to 15.

01:13 If you want to, you can draw a more detailed graph, but i'm just going to draw a sketch here.

01:32 So to plot these points, well, we have the point negative 16 .8, 12 .3 lies on the intersections of the lines x equals negative 16 .8 and y equals 12 .3.

01:54 Now, negative 16 .8 is slightly less than negative 15, and 12 .3 is less than 15.

02:06 Our first point is located somewhere around here.

02:19 Now our next point, 5 .6, 4 .9, is likewise located approximately here.

02:29 5 .6 is a little over 5.

02:32 And 4 .9 is a little bit under 5, which is 1 3rd of 15.

02:43 So we have a plot of our two points.

02:47 Next, in part b, we're asked to find the distance between these points.

02:56 First, let's estimate the distance using our graph from part a.

03:01 So if i draw a line between these two points, a straight line, we want to find the length of this straight line.

03:14 We can approximate this length by recalling the triangle inequality from geometry, which says that the sum of two sides, of the lengths of two sides of a triangle, is always less than, is always, sorry, greater than the length of the third side of the triangle.

03:42 So, for example, if i draw a triangle like this, where the legs are parallel to the x and y axes, this is a right triangle here, then we have that the legs are going to have lengths.

03:58 Let's see, this is going to be 12 .3 minus 4 .9, which i'll just approximate as 12 minus 5 .5.

04:08 Or 7, and this leg is going to have a length of 5 .6 minus negative 16 .7, which i'll approximate as 5 minus 7, negative 17, or 5 plus 17, which is 22.

04:31 We know this third side is going to have a length less than 7 plus 22, some of the other lengths, which is 29.

04:42 So we should expect the length of this third side, and therefore the distance between these two points to be less than 29...

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