THE MIDPOINT FORMULA Recall that point M is the midpoint of segmentPQ if M is between P and Q and PM � MQ. There is a formula for the coordinates ofthe midpoint of a segment in terms of the coordinates of the endpoints. You will

show that this formula is correct in Exercise 41.

412 Chapter 8 Conic Sections

Midpoint and Distance Formulas

MidpointsThe coordinates of the

midpoint are the means

of the coordinates of the

endpoints.

Study TipMidpoint Formula

• Words If a line segment hasendpoints at (x1, y1) and

(x2, y2), then the midpoint

of the segment has

coordinates ��x1 �

2

x2�, �

y1 �

2

y2��.

• Symbols midpoint � ��x1 �

2

x2�, �

y1 �

2

y2��

• Model

y

xO

x1 � x

2y

1 � y2

2 ( , 2 )

(x1, y

1)

(x2, y

2)

• Find the midpoint of a segment on the coordinate plane.

• Find the distance between two points on the coordinate plane.

A square grid is superimposed on a map of eastern Nebraska where emergency medical assistance by helicopter is available from both Lincoln and Omaha. Each side of a square represents 10 miles. You can use the formulas in this lesson to determine whether the site of an emergency is closer to Lincoln or to Omaha.

Find a MidpointLANDSCAPING A landscape designincludes two square flower beds and asprinkler halfway between them. Find thecoordinates of the sprinkler if the origin isat the lower left corner of the grid.

The centers of the flower beds are at (4, 5)and (14, 13). The sprinkler will be at themidpoint of the segment joining these points.

��x1 �

2

x2�, �

y1 �

2

y2�� � ��4 �

214

�, �5 �

213

��� ��

128�, �

128�� or (9, 9)

The sprinkler will have coordinates (9, 9).

Lawn Flowers

Ground cover

FlowersPaved Patio

Example 1Example 1

are the Midpoint and

Distance Formulas used

in emergency medicine?

are the Midpoint and

Distance Formulas used

in emergency medicine?

Mis souri Riv

er

92

7780

OMAHAFremont

Wahoo

Nebraska

LINCOLN

Distance Formula• Words The distance between two

points with coordinates (x1, y1) and (x2, y2) is given by

d � �(x2 ��x1)2 �� (y2 �� y1)2�.

• Model

(x1, y

1)

(x2, y

2)y

x

Od � �(x

2 � x

1)2 � (y

2 � y

1)2

Find the Distance Between Two PointsWhat is the distance between A(�3, 6) and B(4, �4)?

d � �(x2 ��x1)2�� (y2 �� y1)2� Distance Formula

� �[4 � (��3)]2�� (�4� � 6)2� Let (x1, y1) � (�3, 6) and (x2, y2) � (4, �4).

� �72� (��10)2� Subtract.

� �49 � 1�00� or �149� Simplify.

The distance between the points is �149� units.

Example 2Example 2

THE DISTANCE FORMULA Recall that the distance between two points on anumber line whose coordinates are a and b is |a � b| or |b � a|. You can use thisfact and the Pythagorean Theorem to derive a formula for the distance between twopoints on a coordinate plane.

Suppose (x1, y1) and (x2, y2) name two points. Draw a right triangle with vertices at these pointsand the point (x1, y2). The lengths of the legs of theright triangle are |x2 � x1| and |y2 � y1|. Let drepresent the distance between (x1, y1) and (x2, y2).Now use the Pythagorean Theorem.

c2� a2

� b2 Pythagorean Theorem

d2� x2 � x12

� y2 � y12 Replace c with d, a withx2 � x1, and b with y2 � y1.

d2� (x2 � x1)2

� (y2 � y1)2 x2 � x12� (x2 � x1)2; y2 � y12

� (y2 � y1)2

d � �(x2 ��x1)2�� (y2 �� y1)2� Find the nonnegative square root of each side.

(x1, y

1)

d

(x1, y

2) (x

2, y

2)

y

xO

|y2 � y

1|

|x2 � x

1|

DistanceIn mathematics, distances

are always positive.

Study Tip

Investigating Slope-Intercept Form 417(continued on the next page)

Algebra Activity Midpoint and Distance Formulas in Three Dimensions 417

A Follow-Up of Lesson 8-1

You can derive a formula for distance in three-dimensional space. It may seem thatthe formula would involve a cube root, but it actually involves a square root, similarto the formula in two dimensions.

Suppose (x1, y1, z1) and (x2, y2, z2) name two points in space. Draw the rectangular box that has oppositevertices at these points. The dimensions of the box are

x2 � x1, y2 � y1, and z2 � z1. Let a be the

length of a diagonal of the bottom of the box. By the

Pythagorean Theorem, a2� x2 � x12

� y2 � y12.

To find the distance d between (x1, y1, z1) and (x2, y2, z2), apply the Pythagorean Theorem to theright triangle whose legs are a diagonal of the bottom of the box and a vertical edge of the box.

d2� a2

� z2 � z12 Pythagorean Theorem

d2� x2 � x12

� y2 � y12� z2 � z12 a2

� x2 � x12� y2 � y12

d2� (x2 � x1)2

� (y2 � y1)2� (z2 � z1)2 x2 � x12

� (x2 � x1)2, and so on

d � �(x2 ��x1)2�� (y2 �� y1)2

�� (z2 �� z1)2� Take the square root of each side.

The distance d between the points with coordinates (x1, y1, z1) and (x2, y2, z2) is given

by the formula d � �(x2 ��x1)2�� (y2 �� y1)2

�� (z2 �� z1)2�.

z

y

x

O

(x1, y1, z1)

(x2, y2, z2)

a

d

Midpoint and Distance Formulas in Three Dimensions

Example 1Find the distance between (2, 0, �3) and (4, 2, 9).

d � �(x2 ��x1)2�� (y2 �� y1)2

�� (z2 �� z1)2� Distance Formula

� �(4 � 2�)2� (2� � 0)2� � [9 �� (�3)]�2�

� �22� 2�2

� 12�2�

� �152� or 2�38�

The distance is 2�38� or about 12.33 units.

(x1, y1, z1) � (2, 0, �3)(x2, y2, z2) � (4, 2, 9)

In three dimensions, the midpoint of the segment with coordinates (x1, y1, z1) and

(x2, y2, z2) has coordinates ��x1 �

2

x2�, �

y1 �

2

y2�, �

z1 �

2

z2��. Notice how similar this is to

the Midpoint Formula in two dimensions.

(2, 0, �3)

(4, 2, 9)

z

y

x

O 22

46

8

�2

�2�4

�6�8

�4

�6

�8

�4�6�8 4 6 8

2

6

8

Conic Sections

can you use a flashlight to make conic sections?can you use a flashlight to make conic sections?

Recall that parabolas, circles, ellipses, and hyperbolas are called conic sectionsbecause they are the cross sections formed when a double cone is sliced by a plane. You can use a flashlight and a flat surface to make patterns in theshapes of conic sections.

hyperbolaellipsecircleparabola

Reading MathIn this lesson, the word

ellipse means an ellipse

that is not a circle.

Standard Form of Conic Sections

Conic Section Standard Form of Equation

Parabola y � a(x � h)2� k or x � a(y � k)2

� h

Circle (x � h)2� (y � k)2

� r2

Ellipse �(x �

a2h)2� � �

(y �

b2

k)2

� � 1 or �(y �

a2

k)2

� � �(x �

b2h)2� � 1, a � b

Hyperbola �(x �

a2h)2� � �

(y �

b2

k)2

� � 1 or �(y �

a2

k)2

� � �(x �

b2h)2� � 1

Study Tip

STANDARD FORM The equation of any conic section can be written in theform of the general quadratic equation Ax2

� Bxy � Cy2� Dx � Ey � F � 0, where

A, B, and C are not all zero. If you are given an equation in this general form, youcan complete the square to write the equation in one of the standard forms you havelearned.

• Write equations of conic sections in standard form.

• Identify conic sections from their equations.

Rewrite an Equation of a Conic SectionWrite the equation x2

� 4y2� 6x � 7 � 0 in standard form. State whether the graph

of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation.

Write the equation in standard form.

x2� 4y2

� 6x � 7 � 0 Original equation

x2� 6x � � � 4y2

� 7 � � Isolate terms.

x2� 6x � 9 � 4y2

� 7 � 9 Complete the square.

(x � 3)2� 4y2

� 16 x2� 6x � 9 � (x � 3)2

�(x

1�

63)2� � �

y

4

2

� � 1 Divide each side by 16.

The graph of the equation is an ellipse with its center at (3, 0).

y

xO

� � 1(x � 3)2

16

y2

4

Example 1Example 1

Lesson 8-6 Conic Sections 449www.algebra2.com/extra_examples

Identifying Conic Sections

Conic Section Relationship of A and C

Parabola A � 0 or C � 0, but not both.

Circle A � C

Ellipse A and C have the same sign and A � C.

Hyperbola A and C have opposite signs.

IDENTIFY CONIC SECTIONS Instead of writing the equation in standardform, you can determine what type of conic section an equation of the form Ax2

�

Bxy � Cy2� Dx � Ey � F � 0, where B � 0, represents by looking at A and C.

Analyze an Equation of a Conic SectionWithout writing the equation in standard form, state whether the graph of eachequation is a parabola, circle, ellipse, or hyperbola.

a. y2� 2x2

� 4x � 4y � 4 � 0

A � �2 and C � 1. Since A and C have opposite signs, the graph is a hyperbola.

b. 4x2� 4y2

� 20x � 12y � 30 � 0

A � 4 and C � 4. Since A � C, the graph is a circle.

c. y2� 3x � 6y � 12 � 0

C � 1. Since there is no x2 term, A � 0. The graph is a parabola.

Example 2Example 2