Geometry Chapter 1 · Midpoint and Distance Formulas Objective: Students will be able to find the...
Transcript of Geometry Chapter 1 · Midpoint and Distance Formulas Objective: Students will be able to find the...
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