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TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so...
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Transcript of TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so...
![Page 1: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/1.jpg)
TRIANGLES!!
Today, you will learn about various characteristics and properties of triangles … so
get your pencils ready because there are A LOT of
definitions!
![Page 2: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/2.jpg)
Bisectors of a Triangle
Perpendicular bisector of a triangle: a line/ray/segment that is perpendicular to a side of the triangle at the midpoint of the side.
![Page 3: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/3.jpg)
Perpendicular Bisectors of different triangles
![Page 4: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/4.jpg)
Concurrency
When three or more lines intersect in the same point, they are called concurrent lines.
The point of intersection of the lines is called the point of concurrency.
![Page 5: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/5.jpg)
Thm:Concurrency of Perpendicular Bisectors of a
TriangleThe perpendicular
bisectors of a triangle intersect at a point that is equidistant from the vertices of a the triangle.
*perpendicular bisectors*distance from circumcenter to
vertex (all congruent)
![Page 6: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/6.jpg)
The point of concurrency of the perpendicular bisectors of a triangle
is called the circumcenter.
![Page 7: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/7.jpg)
According to our thm., AP = BP = CP.
(Circumcenter is equidistant from vertices)
A
B
C
P
![Page 8: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/8.jpg)
Bisectors of a Triangle
Angle bisector of a triangle: a line/ray/segment that bisects the angle of a triangle.
![Page 9: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/9.jpg)
Thm:Concurrency of Angle Bisectors of a Triangle
The angle bisectors of a triangle intersect at a point that is equidistant from the sides of a the triangle.
*angle bisectors
*distance from incenter to sides (all congruent)
![Page 10: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/10.jpg)
The point of concurrency of angle bisectors of a triangle is called the
incenter.
![Page 11: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/11.jpg)
According to our thm., KM = JM = ML.
(Incenter is equidistant from sides)
J
L
MK
![Page 12: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/12.jpg)
Example
a. Assuming there is a point at A,B, and C, which point is the best spot to place a sprinkler to water plants located at X,Y, and Z?
b. If the sprinkler covers a circular region with a radius of 15 in., will water reach all three plants?
Z
XY
C
B
A
10 in.
![Page 13: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/13.jpg)
Medians of a triangle
Median of a triangle: a segment that starts at the vertex of a triangle and ends at the midpoint of the opposite side.
![Page 14: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/14.jpg)
Thm: Concurrency of Medians of a Triangle
The medians of a triangle intersect at a point such that the distance from the vertex to that point is two-thirds of the length of the entire median within the triangle.
* Medians of the triangle
![Page 15: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/15.jpg)
The point of concurrency of medians of a triangle is called the centroid.
![Page 16: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/16.jpg)
According to our thm., PM = (2/3)PB. (This holds for all 3 medians.)
P
Q
R
B
C
A
M
![Page 17: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/17.jpg)
Altitudes of a Triangle
Altitude of a triange: a line that starts at the vertex and ends at the opposite side (OR line that contains the opposite side) such that is is perpendicular to the opposite side.
![Page 18: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/18.jpg)
Altitudes of different triangles
![Page 19: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/19.jpg)
Concurrency of Altitudes of a Triangle
The altitudes of a triangle intersect at a point called the orthocenter.
* Altitudes of the triangle
![Page 20: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/20.jpg)
Example
C is the centroid of GHJ and CM = 8.
a. Find HM
b. Find CH
H
JG
C
![Page 21: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/21.jpg)
Example
Where is the orthocenter located in ABC if:
a. m of angle A = m of angle B = m of angle C
b. m of angle A = m of angle B = 45 degrees
c. m of angle A = 110 degrees
![Page 22: TRIANGLES!! Today, you will learn about various characteristics and properties of triangles … so get your pencils ready because there are A LOT of definitions!](https://reader035.fdocuments.us/reader035/viewer/2022070414/5697c01f1a28abf838cd1dc4/html5/thumbnails/22.jpg)
Checkpoint!!
1. What is an altitude?
2. What is the pt of concurrency of altitudes?
3. What is a perpendicular bisector?
4. What is the pt of concurrency of per. bis.?
5. What is a median?
6. What is the pt of concurrency of medians?
7. What is an angle bisector?
8. What is the pt of concurrency of angle bisectors?