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1-2 Measuring Segments. Objectives. Use length and midpoint of a segment. Apply distance and midpoint formula. Vocabulary. coordinate midpoint distance bisect length segment bisector congruent segments.

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1-2 Measuring Segments

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1 2 measuring segments

1-2 Measuring Segments

Objectives

Use length and midpoint of a segment.

Apply distance and midpoint formula.


1 2 measuring segments

Vocabulary

coordinatemidpoint

distancebisect

lengthsegmentbisectorcongruent segments


1 2 measuring segments

A point corresponds to one and only one number (or coordinate) on a number line.


1 2 measuring segments

A

B

AB = |a – b| or |b - a|

a

b

Distance (length): the absolute value of the difference of the coordinates.


1 2 measuring segments

Example 1: Finding the Length of a Segment

Find each length.

A. BC

B. AC

BC = |1 – 3|

AC = |–2 – 3|

= |– 2|

= |– 5|

= 2

= 5


1 2 measuring segments

Point B is betweentwo points A and Cif and only if all three points are collinear and AB + BC = AC.

A


1 2 measuring segments

bisect: cut in half; divide into 2 congruent parts.

midpoint: the point that bisects, or divides, the segment into two congruent segments


1 2 measuring segments

4x + 6 = 7x - 9

+9 +9

4x + 15 = 7x

-4x -4x

15 = 3x

3

5 = x


1 2 measuring segments

It’s Mr. Jam-is-on Time!


Recap

Recap!

1. M is between N and O. MO = 15, and MN = 7.6. Find NO.

2.S is the midpoint of TV, TS = 4x – 7, and SV = 5x – 15. Find TS, SV, and TV.

3. LH bisects GK at M. GM =2x + 6, and

GK = 24.Find x.


1 2 measuring segments

1. M is between N and O. MO = 15, and MN = 7.6. Find NO.


1 2 measuring segments

2. S is the midpoint of TV, TS = 4x – 7, and

SV = 5x – 15. Find TS, SV, and TV.


1 2 measuring segments

3.LH bisects GK at M. GM =2x + 6, and

GK = 24.Find x.


1 6 midpoint and distance in the coordinate plane

1-6 Midpoint and Distance in the Coordinate Plane


Vocabulary

Vocabulary

  • Coordinate plane: a plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y-axis.

y-axis

The location, or coordinates, of a point is given by an ordered pair (x, y).

II

I

x-axis

III

IV


Midpoint formula

Midpoint Formula

The midpoint M of a AB with endpoints

A(x1, y1) and B(x2, y2) is found by


Example

Example

Find the midpoint of GH with endpoints G(1, 2) and H(7, 6).


Example1

Example

M(3, -1) is the midpoint of CD and C has coordinates (1, 4).

Find the coordinates of D.


Distance formula

Distance Formula

The distance d between points A(x1, y1) and B(x2, y2) is


Example2

Example

Use the Distance Formula to find the distance between A(1, 2) and B(7, 6).


Pythagorean theorem

Pythagorean Theorem

In a right triangle,

a2 + b2 = c2

c is the hypotenuse (longest side, opposite the right angle)

a and b are the legs (shorter sides that form the right angle)


Example3

Example

Use the Pythagorean Theorem to find the distance between J(2, 1) and K(7 ,7).


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