1 MEDIANS ARE PROPORTIONAL ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PERIMETERS...
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Transcript of 1 MEDIANS ARE PROPORTIONAL ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PERIMETERS...
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1
MEDIANS AREPROPORTIONAL
ALTITUDES AREPROPORTIONAL
ANGLE BISECTORSARE PROPORTIONAL
PERIMETERS AREPROPORTIONAL
ANGLE BISECTORSFORM PROPOR. SEG.
PROBLEM 6
PROBLEM 7
PROBLEM 8
PROBLEM 9
STANDARDS 4 and 5
END SHOW
SPECIAL SEGMENTS
IN A TRIANGLE
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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Standard 4:
Students prove basic theorems involving congruence and similarity.
Los estudiantes prueban teoremas básicos que involucran congruencia y semejanza.
Standard 5:
Students prove triangles are congruent or similar and are able to use the concept of corresponding parts of congruent triangles.
Los estudiantes prueban que triángulos son congruentes o semejantes y son capaces de usar el concepto de partes correspondientes de triángulos congruentes.
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
![Page 3: 1 MEDIANS ARE PROPORTIONAL ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PERIMETERS ARE PROPORTIONAL ANGLE BISECTORS FORM PROPOR. SEG. PROBLEM.](https://reader033.fdocuments.us/reader033/viewer/2022061606/55162062550346a2308b596c/html5/thumbnails/3.jpg)
3STANDARDS 4 and 5
ALTITUDE:
A segment from a vertex of a triangle perpendicular to the line containing the opposite side.
ALTURA:
Un segmento desde el vértice de un triángulo perpendicular a la línea conteniendo el lado opuesto.
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
![Page 4: 1 MEDIANS ARE PROPORTIONAL ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PERIMETERS ARE PROPORTIONAL ANGLE BISECTORS FORM PROPOR. SEG. PROBLEM.](https://reader033.fdocuments.us/reader033/viewer/2022061606/55162062550346a2308b596c/html5/thumbnails/4.jpg)
4STANDARDS 4 and 5
PERPENDICULAR BISECTOR:
A line or segment that passes through the midpoint of a side of a triangle and is perpendicular to that side..
BISECTRIZ PERPENDICULAR:
Una línea o segmento que pasa a través del punto medio de un lado de un triángulo y es perpendicular a ese lado.
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
![Page 5: 1 MEDIANS ARE PROPORTIONAL ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PERIMETERS ARE PROPORTIONAL ANGLE BISECTORS FORM PROPOR. SEG. PROBLEM.](https://reader033.fdocuments.us/reader033/viewer/2022061606/55162062550346a2308b596c/html5/thumbnails/5.jpg)
5STANDARDS 4 and 5
MEDIAN:
A segment that connects a vertex of a triangle to the midpoint of the opposite side.
MEDIANA:
Un segmento que conecta un vertice de un triángulo a el punto medio del lado opuesto.
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
![Page 6: 1 MEDIANS ARE PROPORTIONAL ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PERIMETERS ARE PROPORTIONAL ANGLE BISECTORS FORM PROPOR. SEG. PROBLEM.](https://reader033.fdocuments.us/reader033/viewer/2022061606/55162062550346a2308b596c/html5/thumbnails/6.jpg)
6STANDARDS 4 and 5
ANGLE BISECTOR:
A segment from a vertex to the opposite side that bisects (divides in two equal parts) the angle of the triangle.
BISECTRIZ ANGULAR:
Un segmento desde un vértice a el lado opuesto que biseca (divide en dos partes iguales) el ángulo de el triángulo.
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
![Page 7: 1 MEDIANS ARE PROPORTIONAL ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PERIMETERS ARE PROPORTIONAL ANGLE BISECTORS FORM PROPOR. SEG. PROBLEM.](https://reader033.fdocuments.us/reader033/viewer/2022061606/55162062550346a2308b596c/html5/thumbnails/7.jpg)
7STANDARDS 4 and 5
Draw an isosceles triangle and draw an altitude, a perpendicular bisector, a median and an angle bisector from the vertex angle. What do you observe?
Altitude
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
![Page 8: 1 MEDIANS ARE PROPORTIONAL ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PERIMETERS ARE PROPORTIONAL ANGLE BISECTORS FORM PROPOR. SEG. PROBLEM.](https://reader033.fdocuments.us/reader033/viewer/2022061606/55162062550346a2308b596c/html5/thumbnails/8.jpg)
8STANDARDS 4 and 5
Draw an isosceles triangle and draw an altitude, a perpendicular bisector, a median and an angle bisector from the vertex angle. What do you observe?
Perpendicualr Bisector
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
![Page 9: 1 MEDIANS ARE PROPORTIONAL ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PERIMETERS ARE PROPORTIONAL ANGLE BISECTORS FORM PROPOR. SEG. PROBLEM.](https://reader033.fdocuments.us/reader033/viewer/2022061606/55162062550346a2308b596c/html5/thumbnails/9.jpg)
9STANDARDS 4 and 5
Draw an isosceles triangle and draw an altitude, a perpendicular bisector, a median and an angle bisector from the vertex angle. What do you observe?
Median
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
![Page 10: 1 MEDIANS ARE PROPORTIONAL ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PERIMETERS ARE PROPORTIONAL ANGLE BISECTORS FORM PROPOR. SEG. PROBLEM.](https://reader033.fdocuments.us/reader033/viewer/2022061606/55162062550346a2308b596c/html5/thumbnails/10.jpg)
10STANDARDS 4 and 5
Draw an isosceles triangle and draw an altitude, a perpendicular bisector, a median and an angle bisector from the vertex angle. What do you observe?
Angle Bisector
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
![Page 11: 1 MEDIANS ARE PROPORTIONAL ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PERIMETERS ARE PROPORTIONAL ANGLE BISECTORS FORM PROPOR. SEG. PROBLEM.](https://reader033.fdocuments.us/reader033/viewer/2022061606/55162062550346a2308b596c/html5/thumbnails/11.jpg)
11STANDARDS 4 and 5
Draw an isosceles triangle and draw an altitude, a perpendicular bisector, a median and an angle bisector from the vertex angle. What do you observe?
Altitude
Perpendicualr Bisector
MedianAngle Bisector
Lets put them all together
So, all of them occupy the same Geometric Space in the triangle!
Could you do something similar with an equilateral triangle in all the vertices?
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
![Page 12: 1 MEDIANS ARE PROPORTIONAL ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PERIMETERS ARE PROPORTIONAL ANGLE BISECTORS FORM PROPOR. SEG. PROBLEM.](https://reader033.fdocuments.us/reader033/viewer/2022061606/55162062550346a2308b596c/html5/thumbnails/12.jpg)
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A
BC
K
L
M
KLAB
MK
CALMBC
STANDARDS 4 and 5
CAB MKL
KLAB
MKCA
LMBC
OR
REVIEW:
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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A
BC
K
L
M
ABKL
CA
MKBCLM
N
KNDAD
In simlar triangles MEDIANS are proportional to sides:
STANDARDS 4 and 5
KLAB
MKCA
LMBC
KNAD
OR
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
![Page 14: 1 MEDIANS ARE PROPORTIONAL ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PERIMETERS ARE PROPORTIONAL ANGLE BISECTORS FORM PROPOR. SEG. PROBLEM.](https://reader033.fdocuments.us/reader033/viewer/2022061606/55162062550346a2308b596c/html5/thumbnails/14.jpg)
14
J
I KS
J
UT R
TU
IK
STANDARDS 4 and 5
=JS
6 10
30=
IJK TJU, R is the midpoint of TU and S is the midpoint of IK. TU= 10, JR= 6, IK = 30. Find JS.
S
R
J
UT
I K
JS
JR
(6) (6)
=(30)(6)
10JS
= 180
10JS
JS=18
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
![Page 15: 1 MEDIANS ARE PROPORTIONAL ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PERIMETERS ARE PROPORTIONAL ANGLE BISECTORS FORM PROPOR. SEG. PROBLEM.](https://reader033.fdocuments.us/reader033/viewer/2022061606/55162062550346a2308b596c/html5/thumbnails/15.jpg)
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A
BC
K
L
M
In similar triangles ALTITUDES are proportional to sides:
ABKL
CA
MK
BCLM
O
KOEAE
STANDARDS 4 and 5
KLAB
MK
CA
LMBC
KOAE
OR
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
![Page 16: 1 MEDIANS ARE PROPORTIONAL ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PERIMETERS ARE PROPORTIONAL ANGLE BISECTORS FORM PROPOR. SEG. PROBLEM.](https://reader033.fdocuments.us/reader033/viewer/2022061606/55162062550346a2308b596c/html5/thumbnails/16.jpg)
16
FAF
P
KP
A
BC
K
L
M
In similar triangles ANGLE BISECTORS are proportional to sides:
ABKL
CA
MK
BC
LM
STANDARDS 4 and 5
KLAB
MK
CA
LMBC
KPAF
OR
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
![Page 17: 1 MEDIANS ARE PROPORTIONAL ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PERIMETERS ARE PROPORTIONAL ANGLE BISECTORS FORM PROPOR. SEG. PROBLEM.](https://reader033.fdocuments.us/reader033/viewer/2022061606/55162062550346a2308b596c/html5/thumbnails/17.jpg)
17
JR
JS
JT
JT+TI
STANDARDS 4 and 5
S
R
J
UT
I K
J
I KS
J
UT R
JS is an angle bisector. If TJ = 14, IT=22, and JS= 30, what is the value for JR? Suppose IJK TJU.
14 36
22
14
= JR
30 36
14=
(30) (30)
=(14)(30)
36JR
= 420
36JR
30
JR = 11.7.
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
![Page 18: 1 MEDIANS ARE PROPORTIONAL ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PERIMETERS ARE PROPORTIONAL ANGLE BISECTORS FORM PROPOR. SEG. PROBLEM.](https://reader033.fdocuments.us/reader033/viewer/2022061606/55162062550346a2308b596c/html5/thumbnails/18.jpg)
18
A
BC
K
L
MABKL
CA
MK
BCLM
In similar triangles PERIMETERS proportional to sides:
KL MK LM+ +AB CA BC+ +
STANDARDS 4 and 5
KLAB
MK
CA
LMBC
KL MK LM+ +AB CA BC+ +
OR
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
![Page 19: 1 MEDIANS ARE PROPORTIONAL ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PERIMETERS ARE PROPORTIONAL ANGLE BISECTORS FORM PROPOR. SEG. PROBLEM.](https://reader033.fdocuments.us/reader033/viewer/2022061606/55162062550346a2308b596c/html5/thumbnails/19.jpg)
19
JK
JU JT+JU+TU
PERIMETER IJK
STANDARDS 4 and 5
S
R
J
UT
I K
Given: JT=20, JU=30, TU=70, AND JK=90, what is the perimeter of IJK? Suppose TJU IJK.
J
I KS
J
UT R
=
90
30 20+30+70
PERIMETER IJK=
120
PERIMETER IJK=3(120) (120)
PERIMETER IJK = 360
20 30
7090
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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20
A
B CD
ABACDC
BD
STANDARDS 4 and 5PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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21
IS
SK
JI
JK
STANDARDS 4 and 5
S
R
J
UT
I K
JS is an angle bisector. JI= X+5, JK=X+3, IS=3, and SK=2. Find the value for X. Suppose IJK TJU.
= =
3(X+3) = 2(X+5)
3X + 9 = 2X + 10-9 -9
3X = 2X + 1-2X -2X
X = 1
X+5
X+32
3
X+3X+5
23
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved