SPECIAL SEGMENTS IN A TRIANGLE MEDIANS ARE PROPORTIONAL PROBLEM 6 STANDARDS 4 and 5 MEDIANS ARE PROPORTIONAL PROBLEM 6 ALTITUDES ARE PROPORTIONAL ANGLE BISECTORS ARE PROPORTIONAL PROBLEM 7 PERIMETERS ARE PROPORTIONAL PROBLEM 8 ANGLE BISECTORS FORM PROPOR. SEG. PROBLEM 9 END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Students prove basic theorems involving congruence and similarity. 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

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. STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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. STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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. STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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. STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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 STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Perpendicualr Bisector 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 STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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 STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

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 STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Lets put them all together 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? Lets put them all together Altitude Perpendicualr Bisector Median Angle Bisector 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? STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

REVIEW: CAB MKL A CA AB BC B C LM MK KL KL AB MK CA LM BC K M L OR STANDARDS 4 and 5 REVIEW: CAB MKL A CA AB BC B C LM MK KL KL AB MK CA LM BC OR K M L PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

In simlar triangles MEDIANS are proportional to sides: AD AB KL CA MK BC LM B C KN D K KL AB MK CA LM BC KN AD OR M N L STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

IJK TJU, R is the midpoint of TU and S is the midpoint of IK IJK TJU, R is the midpoint of TU and S is the midpoint of IK. TU= 10, JR= 6, IK = 30. Find JS. JS JR TU IK J I K S U T R S R J U T I K (6) (6) JS 6 10 30 = = = (30)(6) 10 JS = 180 10 JS JS=18 STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

In similar triangles ALTITUDES are proportional to sides: AE AB KL CA MK BC LM B C O KO K KL AB MK CA LM BC KO AE OR M L STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

In similar triangles ANGLE BISECTORS are proportional to sides: F AF AB KL CA MK BC LM B C P KP K KL AB MK CA LM BC KP AF OR M L STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

JS is an angle bisector. If TJ = 14, IT=22, and JS= 30, what is the value for JR? Suppose IJK TJU. 36 14 22 JR JT = (30) (30) JR 30 36 14 = JS JT+TI = (14)(30) 36 JR = 420 36 JR JR = 11.7 . STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

In similar triangles PERIMETERS proportional to sides: AB CA BC + B C K AB KL CA MK BC LM KL MK LM + M KL AB MK CA LM BC + OR L STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

Given: JT=20, JU=30, TU=70, AND JK=90, what is the perimeter of IJK Given: JT=20, JU=30, TU=70, AND JK=90, what is the perimeter of IJK? Suppose TJU IJK. S R J U T I K J I K S U T R 20 30 90 70 JK PERIMETER IJK = JU JT+JU+TU 90 30 20+30+70 PERIMETER IJK = 120 PERIMETER IJK = 3 (120) (120) PERIMETER IJK = 360 STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

A AB AC B C BD DC D STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

JS is an angle bisector. JI= X+5, JK=X+3, IS=3, and SK=2 JS is an angle bisector. JI= X+5, JK=X+3, IS=3, and SK=2. Find the value for X. Suppose IJK TJU. S R J U T I K X+5 X+3 IS JI X+5 X+3 2 3 = 3 2 SK JK 3(X+3) = 2(X+5) 3X + 9 = 2X + 10 -9 -9 3X = 2X + 1 -2X -2X X = 1 STANDARDS 4 and 5 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved