Algebra Review Homework: page 36-37/ 1-8 Coordinate Midpoint Algebra Review Homework: page 36-37/ 1-8

Midpoint of a Segment  

Example 1: Find the coordinates of M, the midpoint of , for G(8, –6) and H(–14, 12). Answer: midpoint (–3, 3)

Your Turn: a. Find the coordinates of the midpoint of for X(–2, 3) and Y(–8, –9).   Answer: (–5, –3)

More About Midpoints You can also find the coordinates of an endpoint of a segment if you know the coordinates of the other endpoint and its midpoint.

Example 2: Find the coordinates of D if E(–6, 4) is the midpoint of and F has coordinates (–5, –3). Let F be in the Midpoint Formula. Write two equations to find the coordinates of D.

Example 2: Solve each equation. Multiply each side by 2. Add 5 to each side. Multiply each side by 2. Add 3 to each side. Answer: The coordinates of D are (–7, 11).

Your Turn: Find the coordinates of R if N(8, –3) is the midpoint of and S has coordinates (–1, 5). Answer: (17, –11)

Example 3: Multiple-Choice Test Item What is the measure of if Q is the midpoint of ? A B 4 C D 9

Example 3: Read the Test Item You know that Q is the midpoint of , and the figure gives algebraic measures for and . You are asked to find the measure of . Solve the Test Item Because Q is the midpoint, you know that . Use this equation and the algebraic measures to find a value for x.

Example 3: Definition of midpoint Distributive Property Subtract 1 from each side. Add 3x to each side. Divide each side by 10.

Example 3: Now substitute for x in the expression for PR. Original measure Simplify. Answer: D

Your Turn: Multiple-Choice Test Item What is the measure of if B is the midpoint of ? A 1 B 3 C 5 D 10 Answer: B

Practice Problems 1. Find the midpoint between the points A(6,4) and B(3,-4). (4.5, 4) B. (3, 0) C. (1.5, 4) D. (4.5, 0) Incorrect. Look at your y-coordinate. You added 4 and (-4) as 8, not 0. Incorrect. Look at your x-coordinate. You divided 9 by 3 instead of 2. Incorrect. Look at your x- and y-coordinates. Check your addition and subtraction. A Correct. Great Job! B

2. The endpoints of a line segment are the points with coordinates (2,1) and (8,9). What are the coordinates of the midpoint of the line segment? (2, 3.5) (3, 4) (5, 5) (10, 10) Incorrect. Double check the coordinates again. Incorrect. Make sure you are adding the coordinates, not subtracting them. Correct. Well done! Incorrect. Don’t forget to divide the sum of the coordinates by 2.

3. You are standing on the point (-4, 6) and your friend is standing on the point (2, 5). You want to walk towards each other and meet halfway. Find the point at which you would meet. (-1. 5.5) (-2, 6) (0, 5.5) (3, 5.5) Correct. Incorrect. Take a look at both coordinates. Make sure you added and subtracted correctly. Incorrect. Take a look at the x-coordinate. Incorrect. Take a look at the x-coordinate. You Your Friend

4. You are given the endpoint of a line segment C(1,2) and its midpoint D(3,1). What is the other endpoint? (5,2) (6,0) C. (5,0) D. (-1,3) Incorrect. This point does not lie in a straight line with the other points. Close! Check the coordinates again. Correct! Incorrect. D, not C should be the midpoint. C D

5. Find the coordinates of the endpoint K if L(-2, -5) is the midpoint of and the coordinates of N are (6, 7). (-10, -17) (10, 17) (-2, 17) (-2, -3) Correct! Incorrect. Check that you have L as the midpoint. Incorrect. Check your x-coordinate and double check that your y-coordinate in the right direction. Incorrect. Look at the placement of this point. Does it make sense for this to be the endpoint in relation to the other two points?

6. If A(2,4) and B(6,4) are the endpoints of a diameter of a circle, find the coordinates of the center of the circle. (3,4) (4,4) (4,5) (8,8) Incorrect. Check your x-coordinate. Make sure you add both the 6 and 2. Correct! Incorrect. Check your y-coordinate. Incorrect. Don’t forget to divide the coordinates by 2 after you add them.

7. You are standing at the point (0,0) and your friend is standing at the point (4,2). You are going to deliver a message to her and you want to meet her halfway. At what point will you meet? (2,0) (2,1) (4,1) (2,2) Incorrect. 2 is halfway between the x-coordinates, but what is halfway between the y-coordinates? Correct! Incorrect. 1 is the correct midpoint in the y-direction, but double check the midpoint in the x-direction. Incorrect. Double check your y-coordinate.

Additional Practice Problems

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

Practice 2: Finding the Coordinates of a Midpoint Find the coordinates of the midpoint M of EF with endpoints E(–2, 3) and F(5, –3).

Practice 3: Finding the Coordinates of an Endpoint M is the midpoint of XY. X has coordinates (2, 7) and M has coordinates (6, 1). Find the coordinates of Y. Step 1 Let the coordinates of Y equal (x, y). Step 2 Use the Midpoint Formula:

Practice 3 Continued Step 3 Find the x and y –coordinates of the endpoint Y by solving each equation. Set the coordinates equal. Multiply both sides by 2. 12 = 2 + x Simplify. 2 = 7 + y – 7 –7 – 2 –2 Subtract. –5 = y 10 = x Simplify. The coordinates of Y are (10, –5).

Practice 4: Finding the Coordinates of an Endpoint S is the midpoint of RT. R has coordinates (–6, –1), and S has coordinates (–1, 1). Find the coordinates of T. Step 1 Let the coordinates of T equal (x, y). Step 2 Use the Midpoint Formula:

Practice 4 Continued Step 3 Find the x-coordinate. Set the coordinates equal. Multiply both sides by 2. –2 = –6 + x Simplify. 2 = –1 + y + 1 + 6 +6 Add. 4 = x Simplify. 3 = y The coordinates of T are (4, 3).