1.3 Use midpoint and distance formulas

You will find lengths of segments in the coordinate plane

Essential question: How do you find the distance and the midpoint between two points in the coordinate plane?

Warm-Up Exercises

In the skateboard design, VW bisects XY at point T, and XT = 39.9 cm. Find XY.

Skateboard

SOLUTION

EXAMPLE 1 Find segment lengths

Point T is the midpoint of XY . So, XT = TY = 39.9 cm.

XY = XT + TY= 39.9 + 39.9= 79.8 cm

Segment Addition PostulateSubstitute.

Add.

Warm-Up Exercises

SOLUTION

EXAMPLE 2 Use algebra with segment lengths

STEP 1 Write and solve an equation. Use the fact that VM = MW.

VM = MW4x – 1 = 3x + 3

x – 1 = 3x = 4

Write equation.

Substitute.

Subtract 3x from each side.Add 1 to each side.

Point M is the midpoint of VW . Find the length of VM .ALGEBRA

Warm-Up ExercisesEXAMPLE 2 Use algebra with segment lengths

STEP 2 Evaluate the expression for VM when x = 4.

VM = 4x – 1 = 4(4) – 1 = 15

So, the length of VM is 15.

Check: Because VM = MW, the length of MW should be 15. If you evaluate the expression for MW, you should find that MW = 15.

MW = 3x + 3 = 3(4) +3 = 15

Warm-Up ExercisesGUIDED PRACTICE for Examples 1 and 2

In Exercises 1 and 2, identify the segment bisectorof PQ . Then find PQ.

1.

343ANSWER MN;

Warm-Up ExercisesGUIDED PRACTICE for Examples 1 and 2

2.

In Exercises 1 and 2, identify the segment bisectorof PQ . Then find PQ.

line l ; 11 57

ANSWER

Warm-Up ExercisesEXAMPLE 3 Use the Midpoint Formula

a. FIND MIDPOINT The endpoints of RS are R(1,–3) and S(4, 2). Find the coordinates of the midpoint M.

Warm-Up ExercisesEXAMPLE 3 Use the Midpoint Formula

252

1 + 4 2

– 3 + 2 2 =, M , – 1M

The coordinates of the midpoint M are 1,–5

2 2

ANSWER

SOLUTION

a. FIND MIDPOINT Use the Midpoint Formula.

Warm-Up ExercisesEXAMPLE 3 Use the Midpoint Formula

b. FIND ENDPOINT The midpoint of JK is M(2, 1). One endpoint is J(1, 4). Find the coordinates of endpoint K.

Warm-Up ExercisesEXAMPLE 3 Use the Midpoint Formula

FIND ENDPOINT Let (x, y) be the coordinates of endpoint K. Use the Midpoint Formula.

STEP 1 Find x.

1+ x 22

=

1 + x = 4

x = 3

STEP 2 Find y.

4+ y 12

=

4 + y = 2

y = – 2

The coordinates of endpoint K are (3, – 2).ANSWER

SOLUTION

Warm-Up ExercisesGUIDED PRACTICE for Example 3

3. The endpoints of AB are A(1, 2) and B(7, 8). Find the coordinates of the midpoint M.

ANSWER (4,5)

ANSWER (– 6, – 8)

4. The midpoint of VW is M(– 1, – 2). One endpoint is W(4, 4). Find the coordinates of endpoint V.

Warm-Up Exercises

SOLUTION

EXAMPLE 4 Standardized Test Practice

Use the Distance Formula. You may find it helpful to draw a diagram.

Warm-Up ExercisesEXAMPLE 4 Standardized Test Practice

Distance Formula

Substitute.

Subtract.

Evaluate powers.

Add.

Use a calculator to approximate the square root.

(x – x ) + (y – y )2 2 2 2 1 1 RS =

[(4 – 2)] + [(–1) –3] 2 2=

(2) + (–4 )2 2=

4+16=

20=

The correct answer is C.ANSWER

4.47~=

Warm-Up ExercisesGUIDED PRACTICE for Example 4

5. In Example 4, does it matter which ordered pair you choose to substitute for (x , y ) and which ordered pair you choose to substitute for (x , y )? Explain.

1

2

1

2

No, when squaring the differences in the coordinates, you get the same answer as long as you choose the x and y values from the same point.

SAMPLE ANSWER

Warm-Up ExercisesGUIDED PRACTICE for Example 4

6. What is the approximate length of AB , with endpoints A(–3, 2) and B(1, –4)?

6.1 units 7.2 units 8.5 units 10.0 units

BANSWER

Warm-Up ExercisesDaily Homework Quiz

1. AB bisects CD at E. If CE = in., Find CD.14

2

2. Point M is the midpoint of XY. Find XM.

ANSWER 17

12

4ANSWER in.

Warm-Up ExercisesDaily Homework Quiz

3. Point M is the midpoint of PQ with endpoints P(2, – 6 ) and Q(– 8, 0). Find the coordinates of M.

ANSWER (–3, –3)

ANSWER (3, –5)

4. The midpoint of GH is M(4, –1). One endpoint is G(5, 3) . Find the coordinates of H.

Warm-Up ExercisesDaily Homework Quiz

5. To find the distance between the swing and the sandbox in his backyard, Darren made a graph and found the coordinates of the swing to be (7, 2) and the coordinates of the sandbox to be (– 3, 8). Find the distance between the swing and the sandbox to the nearest tenth of a unit.

11.7 ANSWER

1.3 Use midpoint and distance formulas

You will find lengths of segments in the coordinate plane

Essential question: How do you find the distance and the midpoint between two points in the coordinate plane?

The midpoint of a segment is the point that divides the segment into two congruent parts.The length of a segment in the coordinate plane is the distance between its endpoints.

To find the distance between the points (a,b) and (c,d) use the distance formula d= (d-b)2+(c-a)2. To find the midpoint, use the midpoint formula (a+c, b+d) 2 2