1.3

Warm UpWarm Up

Lesson QuizLesson Quiz

Lesson PresentationLesson Presentation

Use Midpoint and Distance Formulas

1.3 Warm-Up

1. Find a point between A(–3, 5) and B(7, 5).

2. Find the average of –11 and 5.

ANSWER Sample: (2, 5)

ANSWER –3

1.3 Warm-Up

ANSWER 3

ANSWER 5.48

3. Solve = 5.2

x + 7

4. Find √30 to the nearest hundredth.

1.3 Warm-Up

ANSWER 6.71

5. Find √5 + √20 to the nearest hundredth.

1.3 Example 1

Skateboard

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

SOLUTION

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 Postulate

Substitute.

Add.

1.3 Example 2

SOLUTION

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

VM = MW

4x – 1 = 3x + 3

x – 1 = 3

x = 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

1.3 Example 2

SOLUTION

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

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

1.3 Guided Practice

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

1.

343ANSWER MN;

1.3 Guided Practice

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

2.

line ; 11 57

ANSWER

1.3 Example 3

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

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.

1.3 Example 3

SOLUTION

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

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

STEP 1 Find x.

1+ x 22

=

1 + x = 4x = 3

STEP 2 Find y.

4+ y 12

=

4 + y = 2y = – 2

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

1.3 Guided Practice

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.

1.3 Example 4

SOLUTION

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

1.3 Example 4

SOLUTION

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=

4.47~=

The correct answer is C.ANSWER

1.3 Guided Practice

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

1.3 Guided Practice

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

1.3 Lesson 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.

1.3 Lesson 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.

1.3 Lesson 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