Finding the Midpoint and Length of a Line Segment in Coordinate Geometry, It Is Important to Be Able to Find the Midpoints and the Lengths of Segments

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Finding the Midpoint and Length of a Line Segment In coordinate geometry, it is important to be able to find the midpoints and the lengths of segments.

Example A

a. Find the coordinates of the midpoints of y TP and AR. 10 T(26, 9) b. Compare the length of MN with the sum of the lengths of TR and PA. M 5

P (26, 1) R(12, 0) x 210 25 5 10 15 N A(0, 22)

25 Solution:

a. For the segment whose endpoints are (x1, y1) and (x2, y2), the coordinates of the midpoint  xx11yy are  12, 12. Use the formula to find the midpoints M and N.  2 2 

216(216) 91 212 10 M:  ,  5  ,  5 (26, 5)  2 2   2 2 

 01122 201  5  12 22 5 2 N:  ,   ,  (6, 1)  2 2   2 2 

2 2 b. Use the Distance Formula d 5 ()xx2121()yy212 to find the distance between

(x1, y1) and (x2, y2) . MN 5 (6226)2212(5 (12 )) 5 (1212)2 62 5 144 136 5 180 5 36 ? 5 5 6 5 units

TR 5 (62212)(221290) 5 324 1 81 5 405 5 81 ? 5 5 9 5 units

PA 5 (6220)2212(1 (22 )) 5 36 1 9 5 45 5 3 5 units 951 35 12 5 Notice that the average of 9 5 and 3 5 is 5 5 6 5. In other words, the 2 2 length of MN, which is 6 5 units, is the average of the lengths of TR and PA; or the length of TR and PA together, is twice the length of MN .

Example B

In RST, points V and W are the midpoints of RT and ST. Compare the lengths of RS and VW.

y 10

R(1, 7)

V S(5,1) x 25 5 W

T(23, 23) 25

Step 1: Use R(1, 7) and S(5, 1) to find the length of RS. RS 5 (1 215)22(7 21) 5 16136 5 52 5 2 13 Step 2: Use the Midpoint Formula to the find coordinates of points V and W. 21312137 22 4 V:  ,  5  ,  5 (21, 2)  2 2   2 2 21352131  2 22 W:  ,  5  ,  5 (1, 21)  2 2   2 2  Step 3: Use V(21, 2) and W(1, 21) to find the length of VW.

VW 5 (1221)2212(2 (12 )) 5 491 5 13

Solution: The length of VW is half the length of RS. This actually illustrates the Triangle Midsegment Theorem, which states that the segment joining the midpoints of two sides of a triangle is parallel to the other side of the triangle and is half the length of that side.

PRACTICE Use the diagram shown for Items 1–2.

y 1. Find the midpoints of MT and AH. 5 2 M (0, 3) Both midpoints are (3, 3). 2. Find the lengths of MT and AH. A(23, 0) x Each length is 6 5. 25 5 10

25 H(9, 26)

T(6, 29) 210

Use the diagram shown for Items 3–7.

y 3. Find the coordinates of the midpoint of AC. Do the 15 coordinates satisfy y 5 2x 1 8? Is the midpoint of C(4, 12) AC on the line y 5 2x 1 8? The midpoint is (2, 6). That ordered pair satisfies the 10 equation y 5 2x 1 8, so (2, 6) is on the line. 4. Find the coordinates of the midpoint of BC. P Q x The midpoint is (6, 6). 5 y 5 y 52 5. Find the coordinates of the midpoint of AB.

x 1 R(4, 4) 8 The midpoint is (4, 0).

A(0, 0) B(8, 0) x 0 5 10

6. What is the length of the segment from the midpoint of AB to vertex C? 12 units 7. Find the distance between points C and R. How does that distance compare to the length of the segment from the midpoint of AB to vertex C? 8 units; it is two-thirds of the length of the segment from the midpoint of AB to vertex C.