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1© 2015 College Board. All rights reserved. SpringBoard Geometry, Unit 2

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

a. Find the coordinates of the midpoints of TP and AR.

b. Compare the length of MN with the sum of the lengths of TR and PA.

x

y

5

10

5210 25

25

A (0, 22)

P (26, 1)

M

N

R (12, 0)

T (26, 9)

10 15

Solution: a. For the segment whose endpoints are (x1, y1) and (x2, y2), the coordinates of the midpoint

are

1 1x x y y2

,2

1 2 1 2 . Use the formula to find the midpoints M and N.

M:

2 1 2 16 ( 6)2

, 9 12

5

122

,102

2 5 (26, 5)

N:

0 122

,2 02

1 2 1 5

122,

22

2 5 (6, 21)

b. Use the Distance Formula d 5 2 1 2x x y y( ) ( )2 12

2 12 to find the distance between

(x1, y1) and (x2, y2) .

MN 5 ( 6 6) (5 ( 1))2 22 2 1 2 2 5 ( 12) 62 22 1 5 144 361 5 180 5 ?36 5 5 6 5 units

TR 5 ( 6 12) (9 0)2 22 2 1 2 5 1324 81 5 405 5 ?81 5 5 9 5 units

PA 5 ( 6 0) (1 ( 2))2 22 2 1 2 2 5 136 9 5 45 5 3 5 units

Notice that the average of 9 5 and 3 5 is 19 5 3 52

5 12 52

5 6 5 . In other words, the

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 a

2© 2015 College Board. All rights reserved. SpringBoard Geometry, Unit 2

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

x

y

5

10

525

25

T (23, 23)

R (1, 7)

S (5,1)

W

V

Step 1: Use R(1, 7) and S(5, 1) to find the length of RS.

RS 5 2 1 2(1 5) (7 1)2 2 5 16 361 5 52 5 2 13

Step 2: Use the Midpoint Formula to the find coordinates of points V and W.

V:

3 12

,3 72

2 1 2 1 5

22,42

2 5 (21, 2)

W:

3 52

,3 12

2 1 2 1 5

22,

22

2 5 (1, 21)

Step 3: Use V(21, 2) and W(1, 21) to find the length of VW .

VW 5 2 2 1 2 2( 1 1) (2 ( 1))2 2 5 4 91 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.

ExaMpLE B

Finding the Midpoint and Length of a Line Segment (continued)

3© 2015 College Board. All rights reserved. SpringBoard Geometry, Unit 2

pRaCTICEUse the diagram shown for Items 1–2.

x

y

5

525

25

210

A (23, 0)

H (9, 26)

T (6, 29)

M (0, 3)

10

1.

2.

Find the midpoints of MT and AH .

Both midpoints are (3, 23).

Find the lengths of MT and AH .

each length is 6 5.

Use the diagram shown for Items 3–7.

x

y

0

5

P Q

10

15

5

A (0, 0)

R(4, 4)

B(8, 0)

C(4, 12)

y 5x

10y 5

2x 1

8

3.

4.

5.

Find the coordinates of the midpoint of AC. Do the coordinates satisfy y 5 2x 1 8? Is the midpoint of AC on the line y 5 2x 1 8?

the midpoint is (2, 6). that ordered pair satisfies the equation y 5 2x 1 8, so (2, 6) is on the line.

Find the coordinates of the midpoint of BC.

the midpoint is (6, 6).

Find the coordinates of the midpoint of AB.

the midpoint is (4, 0).

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.

Finding the Midpoint and Length of a Line Segment (continued)