Chapter 1.7 Midpoint and Distance in a Coordinate Plane SWBAT Find the midpoint of a segment. Find the distance between two points in a coordinate plane.

Slope = 𝒓𝒊𝒔𝒆 𝒓𝒖𝒏 = 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏 A coordinate plane is a plane that is divided into four regions by a horizontal line (x-axis) and a vertical line (y-axis) . The location, or coordinates, of a point are given by an ordered pair (x, y). y - axis “Cartesian” Coordinate system Quadrant II (-,+) Quadrant I (+,+) origin (0,0) x - axis Slope = 𝒓𝒊𝒔𝒆 𝒓𝒖𝒏 = 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏 Quadrant III (-,-) Quadrant IV (+,-)

Midpoints = Averages Finding the midpoint between two points on a number line is as simple as finding the average of the points. Add the coordinates and divide by 2. A B -10 -6 -2 2 6 𝑻𝒉𝒆 𝑴𝒊𝒅𝒑𝒐𝒊𝒏𝒕 𝒐𝒇 𝑨𝑩 𝒊𝒔 𝒂𝒕 −𝟏𝟎+𝟔 𝟐 = −𝟒 𝟐 =−𝟐 The same (or at least very similar) process applies to finding midpoints in 2 dimensions as well.

You can find the midpoint of a segment by using the coordinates of its endpoints. Calculate the average of the x-coordinates and the average of the y-coordinates of the endpoints.

To make it easier to picture the problem, plot the segment’s endpoints on a coordinate plane. Helpful Hint

Example 1: Finding the Coordinates of a Midpoint Find the coordinates of the midpoint of PQ with endpoints P(–8, 3) and Q(–2, 7). = (–5, 5)

Check It Out! Example 2 Find the coordinates of the midpoint of EF with endpoints E(–2, 3) and F(5, –3).

Now it’s your turn! Find the coordinates of the midpoint of RS with endpoints R(3, 9) and S(11, –3). Find the coordinates of the midpoint of RS with endpoints R(3, 9) and S(11, –3). Find the coordinates of the midpoint of RS with endpoints R(3, 9) and S(11, –3). The average of the x-coordinates is: (3+11)/2 = 14/2 = 7 The average of the y-coordinates is: (9 + (-3))/2 = 6/2 = 3 The midpoint is at (7, 3)

Finding the Coordinates of an Endpoint Given one Endpoint and the Midpoint M is the midpoint of XY. X has coordinates (2, 7) and M has coordinates (6, 1). Find the coordinates of Y. To do this easily, we can use the idea of the slope of a straight line. Start by drawing a picture of the situation.

Finding the Coordinates of an Endpoint Given one Endpoint and the Midpoint M is the midpoint of XY. X has coordinates (5, 4) and M has coordinates (3, 1). Find the coordinates of Y. From X to M, you move down 3 and left 2. 𝟑−𝟓, 𝟏−𝟒 =(−𝟐, −𝟑) X -2 -3 M For a straight line, slope does not change. To find the coordinates of Y, use that slope again, down 3 and left 2 -2 -3 Y To find the coordinates of Point Y, use the midpoint and your slope. 𝒀 𝒙,𝒚 = 𝟑−𝟐, 𝟏−𝟑 =(𝟏, −𝟐)

Finding the Coordinates of an Endpoint Given one Endpoint and the Midpoint M is the midpoint of PQ. P has coordinates (5, -3) and M has coordinates (1, -1). Find the coordinates of Q. From P to M, you move up 2 and left 4. 𝟏−𝟓, −𝟏−(−𝟑) =(−𝟒, 𝟐) Q For a straight line, slope does not change. To find the coordinates of Q, use that slope again, up 2 and left 4 2 -4 M 2 -4 P To find the coordinates of Point Q, use the midpoint and your slope. Q 𝒙,𝒚 = 𝟏−𝟒, −𝟏+𝟐 =(−𝟑, 𝟏)

The Ruler Postulate can be used to find the distance between two points on a number line. The Distance Formula is used to calculate the distance between two points in a coordinate plane.

Example 5: Using the Distance Formula Find FG and JK. Then determine whether FG  JK. Step 1 Find the coordinates of each point. F(1, 2), G(5, 5), J(–4, 0), K(–1, –3)

Example 5 Continued Step 2 Use the Distance Formula.

Check It Out! Example 6 Find EF and GH. Then determine if EF  GH. Step 1 Find the coordinates of each point. E(–2, 1), F(–5, 5), G(–1, –2), H(3, 1)

Check It Out! Example 6 Continued Step 2 Use the Distance Formula.

Assignment #8 Pages 54-56 Foundation: 6 – 33 divisible by 3 Core: 36 – 42 divisible by 3 Challenge: 60