Chapter 5 Properties of Triangles
Transcript of Chapter 5 Properties of Triangles
![Page 1: Chapter 5 Properties of Triangles](https://reader030.fdocuments.us/reader030/viewer/2022012710/61aa408dd8ac31420751735a/html5/thumbnails/1.jpg)
1
Chapter 5Properties of Triangles
![Page 2: Chapter 5 Properties of Triangles](https://reader030.fdocuments.us/reader030/viewer/2022012710/61aa408dd8ac31420751735a/html5/thumbnails/2.jpg)
2
Learning Targets (KRSP):I can……
1. Identify a perpendicular and angle bisector, with the knowledge of equidistant, create a drawing that represents it, then solve problems involving a perpendicular bisector including proving the perpendicular bisector theorem;
2. Recognize and demonstrate perpendicular and angle bisectors in a triangle and label their points of concurrency as circumcenters and incenters respectively;
3. Solve problems with triangles involving perpendicular and angle bisectors;4. Create medians and altitudes of a triangle and label their points of
concurrency as centroids and orthocenters respectively;5. Construct midsegments of a triangle on a graph;6. Apply the triangle inequality to determine characteristics of a triangle.
![Page 3: Chapter 5 Properties of Triangles](https://reader030.fdocuments.us/reader030/viewer/2022012710/61aa408dd8ac31420751735a/html5/thumbnails/3.jpg)
3
Learning Target #1Identify a perpendicular bisector, with the knowledge of equidistant, create a drawing that represents it, then solve problems involving a perpendicular bisector
![Page 4: Chapter 5 Properties of Triangles](https://reader030.fdocuments.us/reader030/viewer/2022012710/61aa408dd8ac31420751735a/html5/thumbnails/4.jpg)
4
Perpendicular Bisector
A B
equidistant
![Page 5: Chapter 5 Properties of Triangles](https://reader030.fdocuments.us/reader030/viewer/2022012710/61aa408dd8ac31420751735a/html5/thumbnails/5.jpg)
5
Perpendicular Bisector Theorem
BA
C
M
Prove it:Given: CM is bisector of ABProve: CA = CB
Statements Reasons1. CM is bisector of AB 1. Given2. AM = BM 2. def. of bisector3. ___________________ 3. def. of ≅ segments4. ___________________ 4. 2 lines, 4 rt. 5. CMA ≅ CMB 5. ___________________6. ___________________ 6. ___________________7. ΔCMA ≅ ΔCMB 7. ___________________8. CA ≅ CB 8. ___________________9. ___________________ 9. ___________________
![Page 6: Chapter 5 Properties of Triangles](https://reader030.fdocuments.us/reader030/viewer/2022012710/61aa408dd8ac31420751735a/html5/thumbnails/6.jpg)
6
12
12
T
M N
S
Q
ex: In the diagram shown, MN is the bisector of ST a. What segments are equal?
b. Explain why Q is on MN.
![Page 7: Chapter 5 Properties of Triangles](https://reader030.fdocuments.us/reader030/viewer/2022012710/61aa408dd8ac31420751735a/html5/thumbnails/7.jpg)
7
Recall: Angle Bisector
Angle Bisector Theorem
D C
A
B
![Page 8: Chapter 5 Properties of Triangles](https://reader030.fdocuments.us/reader030/viewer/2022012710/61aa408dd8ac31420751735a/html5/thumbnails/8.jpg)
8
Learning Target #2Recognize and demonstrate perpendicular and
angle bisectors in a triangle and label their points of concurrency as circumcenters and incenters
respectively
![Page 9: Chapter 5 Properties of Triangles](https://reader030.fdocuments.us/reader030/viewer/2022012710/61aa408dd8ac31420751735a/html5/thumbnails/9.jpg)
9
Triangle ActivityGet into 4 groups
With your triangle, fold the bisectors of each side (2 members fold the angle bisectors, 2 members fold the perpendicular bisectors).
Then trace the bisectors with your straight edge. What do you observe?
![Page 10: Chapter 5 Properties of Triangles](https://reader030.fdocuments.us/reader030/viewer/2022012710/61aa408dd8ac31420751735a/html5/thumbnails/10.jpg)
10
ASSIGNMENTp. 267 #313, 1626,
3335
![Page 11: Chapter 5 Properties of Triangles](https://reader030.fdocuments.us/reader030/viewer/2022012710/61aa408dd8ac31420751735a/html5/thumbnails/11.jpg)
11
Concurrency
Concurrent lines
Point of Concurrency (P.O.C.)
**THE THREE BISECTORS OF A TRIANGLE ARE CONCURRENT!
P
P
P
Circumcenter
![Page 12: Chapter 5 Properties of Triangles](https://reader030.fdocuments.us/reader030/viewer/2022012710/61aa408dd8ac31420751735a/html5/thumbnails/12.jpg)
12
P
AB
C
The bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle
PA = PB = PC
Circumscribed
= bisector
= radius of P
![Page 13: Chapter 5 Properties of Triangles](https://reader030.fdocuments.us/reader030/viewer/2022012710/61aa408dd8ac31420751735a/html5/thumbnails/13.jpg)
13
Angle Bisector of a Triangleangle bisectors are concurrent Incenter
P
![Page 14: Chapter 5 Properties of Triangles](https://reader030.fdocuments.us/reader030/viewer/2022012710/61aa408dd8ac31420751735a/html5/thumbnails/14.jpg)
14
The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle
P
B
A
C
D
E
F
Inscribed
PD = PE = PF
= angle bisector= radius of P
![Page 15: Chapter 5 Properties of Triangles](https://reader030.fdocuments.us/reader030/viewer/2022012710/61aa408dd8ac31420751735a/html5/thumbnails/15.jpg)
15
Learning Target #3Solve problems with triangles
involving perpendicular and angle bisectors
![Page 16: Chapter 5 Properties of Triangles](https://reader030.fdocuments.us/reader030/viewer/2022012710/61aa408dd8ac31420751735a/html5/thumbnails/16.jpg)
16
The perpendicular bisectors of ΔABC meet at point P.
Which segments are congruent?
Find PC.
P
AB
C
85
7
D
E F
Find BD.
Find AB.
![Page 17: Chapter 5 Properties of Triangles](https://reader030.fdocuments.us/reader030/viewer/2022012710/61aa408dd8ac31420751735a/html5/thumbnails/17.jpg)
17
The angle bisectors of ΔABC meet at point P.
Which segments are congruent?
Find PF.
Find BP.
P
B
A
C
D
E
F
6
8
15
Find AF.
![Page 18: Chapter 5 Properties of Triangles](https://reader030.fdocuments.us/reader030/viewer/2022012710/61aa408dd8ac31420751735a/html5/thumbnails/18.jpg)
18
The angle bisectors of ΔBRO intersect at point M.
Which segments are congruent?
Find ME.
B
R
O
M
T
A
E
54
![Page 19: Chapter 5 Properties of Triangles](https://reader030.fdocuments.us/reader030/viewer/2022012710/61aa408dd8ac31420751735a/html5/thumbnails/19.jpg)
19
ASSIGNMENTp. 275 #3, 4, 1017