Geometry Chapter 11-3: Midpoint and Distance Formulas

Warm-Up

1.) Find the average of −11 and 5

2.) Find the average of −10 and 20

3.) Find 144 to the nearest hundredth

4.) Find 30 to the nearest hundredth

Sol: −11+5

2=

−6

2= −3

Sol: −10+20

2=

10

2= 5

Sol: 144 = 12

Sol: 30 ≈ 5.48

Midpoint and Distance Formulas

Objective: Students will be able to find the lengths of segments in the coordinate plane using the Distance and Midpoint Formulas.

Agenda:

•Midpoint/Segment Bisector

•Midpoint Formula

• The Distance Formula

Midpoints and Bisectors

• The Midpoint of a segment is the point that divides the segment into two congruent segments.

A BM

𝑴 is the midpoint of 𝑨𝑩.

So, 𝑨𝑴 = 𝑴𝑩 and 𝑨𝑴 ≅ 𝑴𝑩.

Midpoints and Bisectors

• A Segment Bisector is a point, ray, line, line segment, or plane that intersects the line segment at its midpoint.

D

C

A B

M

𝑪𝑫 is a segment bisector of 𝑨𝑩.

So, 𝑨𝑴 = 𝑴𝑩 and 𝑨𝑴 ≅ 𝑴𝑩.

Example 1 (Pg. 15 in the Textbook)

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

X

Y

T

W

VSince point T is the midpoint of 𝑋𝑌, then 𝑋𝑇 = 𝑇𝑌, then 𝑇𝑌 = 39.9

Thus,

𝑋𝑌 = 𝑋𝑇 + 𝑇𝑌

= 39.9 + 39.9

= 𝟕𝟗. 𝟖 𝐜𝐦

Example 2 (Pg. 16 in the Textbook)

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

Step 1: Write the Equation

𝑉𝑀 = 𝑀𝑊 (Why?)

4𝑥 − 1 = 3𝑥 + 3

𝑥 − 1 = 3𝒙 = 𝟒

Step 2: Plug in 𝑥 = 4 for 𝑉𝑀

𝑉𝑀 = 4𝑥 − 1

4(4) − 1

𝟏𝟓

Midpoints of CoordinatesThe Midpoint Formula:

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

Equation:

Given points A (𝑥1, 𝑦1) and B(𝑥2, 𝑦2), the midpoint M is given by the equation

𝑀 =𝑥1 + 𝑥2

2,𝑦1 + 𝑦𝑥

2

Example 3 (Pg. 17 in the Textbook)

a.) The endpoints of 𝑅𝑆 are R (1, −3) and S (4, 2). Find the coordinates of the Midpoint.

𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧:

𝑀 =𝑥1 + 𝑥2

2,𝑦1 + 𝑦𝑥

2

𝑀 =1 + 4

2,−3 + 2

2

𝑀 =5

2,−1

2

Thus, the midpoint is 𝟓

𝟐,−𝟏

𝟐

Example 3 (Pg. 17 in the Textbook)b.) The midpoint of 𝐽𝐾 is M (2, 1). One endpoint is J (1, 4). Find the coordinates of the endpoint K.

𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧:

𝑀 =𝑥1 + 𝑥2

2,𝑦1 + 𝑦𝑥

2

(2, 1) =1 + 𝑥

2,4 + 𝑦

2

Split the Solution:

2 =1 + 𝑥

2

4 = 1 + 𝑥

𝑥 = 3

1 =4 + 𝑦

2

2 = 4 + 𝑦

𝑦 = −2

Thus, The coordinates for point K is (𝟑, −𝟐)

Distances of CoordinatesThe Distance Formula:

Given points A (𝑥1, 𝑦1) and B (𝑥2, 𝑦2), the distance between points A and B is given by the equation

𝐴𝐵 = (𝑥2 − 𝑥1)2+(𝑦2 − 𝑦1)

2

Example 4 (Pg. 18 of the Textbook)What is the approximate length of 𝑅𝑆 with endpoints R (2, 3) and S(4,−1)?

𝑅𝑆 = (𝑥2 − 𝑥1)2+(𝑦2 − 𝑦1)

2

= (4 − 2)2+(−1 − 3)2

= (2)2+(−4)2

= 4 + 16

= 𝟐𝟎 ≈ 𝟒. 𝟒𝟕

Distance Connections

The Distance Formula is based on the Pythagorean Theorem.

Distance Formula: Pythagorean Theorem:

Homework 1-3

• Pg. 19-21 #’s 1, 2, 4, 5, 6, 8, 12-15, 19-22, 23 (EC), 24, 25, 27, 31-33, 41, 42, 48, 49

• EC – Extra Credit