1.3: Use Midpoint and Distance 1.3: Use Midpoint and Distance FormulasFormulas

Objectives:

1.To define midpoint and segment bisector

2.To use the Midpoint and Distance Formulas

3.To construct a segment bisector with a compass and straightedge

Perpendicular BisectorPerpendicular Bisector

1. Draw a segment. Label the endpoints A and B.

Perpendicular BisectorPerpendicular Bisector

2. Using the same compass setting, draw two intersecting arcs through the segment, one centered at A, the other centered at B. Label the intersection points C and D.

Perpendicular BisectorPerpendicular Bisector

3. Draw a line through points C and D.

Perpendicular BisectorPerpendicular Bisector

4. Label the new point of intersection M. Is point is called the midpointmidpoint.

Perpendicular Bisector: Perpendicular Bisector: VideoVideo

Click on the image to watch a video of the construction.

©2001 New York State Regents Exam Prep Center

VocabularyVocabulary

Midpoint Segment bisector

As a group, define each of these without your book. Draw a picture for each word and leave a bit of space for additions and revisions.

MidpointMidpoint

The midpoint of a segmentmidpoint of a segment is the point on the segment that divides, or bisectsbisects, it into two congruent segments.

Segment BisectorSegment Bisector

A segment bisectorsegment bisector is a point, ray, line, line segment, or plane that intersects the segment at its midpoint.

Example 1Example 1

Find DM if M is the midpoint of segment DA, DM = 4x – 1, and MA = 3x + 3.

M

D

A

Example 2: SATExample 2: SAT

In the figure shown, ABCD is a rectangle with BC = 4 and AB = 6. Points P, Q, and R are different points on a line (not shown) that is parallel to AD. Points P and Q are symmetric about line AB and points Q and R are symmetric about line CD. What is PR?

Note: Figure not drawn to scale.

R

Q

D

CB

A

P

Example 3Example 3

Segment OP lies on a real number line with point O at –9 and point P at 3. Where is the midpoint of the segment?

What if the endpoints of segment OP were at x1 and x2?

- 10 -5 50

O P

In the Coordinate PlaneIn the Coordinate Plane

We could extend the previous exercise by putting the segment in the coordinate plane. Now we have two dimensions and two sets of coordinates. Each of these would have to be averaged to find the coordinates of the midpoint.

4

2

-2

-5 5

Midpoints on theCoordinate Plane

Midpoint: (0.00, 1.00)

B: (6.00, 4.00)

A: (-6.00, -2.00)

Midpoint

A

B

The Midpoint FormulaThe Midpoint Formula

If A(x1,y1) and B(x2,y2) are points in a coordinate plane, then the midpoint M of AB has coordinates

2

,2

2121 yyxx

The Midpoint FormulaThe Midpoint Formula

The coordinates of the midpoint of a segment are basically the averages of the x- and y-coordinates of the endpoints

Example 4Example 4

Find the midpoint of the segment with endpoints at (-1, 5) and (3, 8).

After Lunch:After Lunch:

Find the midpoint of the segment with endpoints at (-1, 5) and (3, 8).

The midpoint C of IN has coordinates (4, -3). Find the coordinates of point I if point N is at (10, 2).

Example 6Example 6

Use the Midpoint Formula multiple times to find the coordinates of the points that divide AB into four congruent segments.

A B

Parts of a Right TriangleParts of a Right Triangle

Which segment is the longest in any right triangle?

The Pythagorean TheoremThe Pythagorean Theorem

In a right triangle, if a and b are the lengths of the legs and c is the length of the hypotenuse, then c2 = a2 + b2.

Example 7Example 7

How high up on the wall will a twenty-foot ladder reach if the foot of the ladder is placed five feet from the wall?

The Distance FormulaThe Distance Formula

Sometimes instead of finding a segment’s midpoint, you want to find it’s length. Notice how every non-vertical or non-horizontal segment in the coordinate plane can be turned into the hypotenuse of a right triangle.

Example 8Example 8

Graph AB with A(2, 1) and B(7, 8). Add segments to your drawing to create right triangle ABC. Now use the Pythagorean Theorem to find AB.

Distance FormulaDistance Formula

In the previous problem, you found the length of a segment by connecting it to a right triangle on graph paper and then applying the Pythagorean Theorem. But what if the points are too far apart to be conveniently graphed on a piece of ordinary graph paper? For example, what is the distance between the points (15, 37) and (42, 73)? What we need is a formula!

The Distance FormulaThe Distance Formula

To find the distance between points A and B shown at the right, you can simply count the squares on the side AC and the squares on side BC, then use the Pythagorean Theorem to find AB. But if the distances are too great to count conveniently, there is a simple way to find the lengths. Just use the Ruler Postulate.

8

6

4

2

5

B

CA

The Distance FormulaThe Distance Formula

You can find the horizontal distance subtracting the x-coordinates of points A and B: AC = |7 – 2| = 5. Similarly, to find the vertical distance BC, subtract the y-coordinates of points A and B: BC = |8 – 1| = 7. Now you can use the Pythagorean Theorem to find AB.

8

6

4

2

5

B

CA

Example 9Example 9

Generalize this result and come up with a formula for the distance between any two points (x1, y1) and (x2, y2).

8

6

4

2

5

(x2, y 1)

(x2, y 2)

(x1, y 1)

B

CA

The Distance FormulaThe Distance Formula

If the coordinates of points A and B are (x1, y1) and (x2, y2), then

2122

12 yyxxAB

Example 10Example 10

To the nearest tenth of a unit, what is the approximate length of RS, with endpoints R(3, 1) and S(-1, -5)?

Example 11Example 11

A coordinate grid is placed over a map. City A is located at (-3, 2) and City B is located at (4, 8). If City C is at the midpoint between City A and City B, what is the approximate distance in coordinate units from City A to City C?

Example 12Example 12

Points on a 3-Dimensional coordinate grid can be located with coordinates of the form (x, y, z). Finding the midpoint of a segment or the length of a segment in 3-D is analogous to finding them in 2-D, you just have 3 coordinates with which to work.

Example 12Example 12

Find the midpoint and the length of the segment with endpoints (2, 5, 8) and (-3, 1, 2).