c - Math with Mr. Leon

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Lesson 9.1
In a right triangle, the side opposite the
right angle is called the hypotenuse.
The other two sides are called legs. In
the figure below, a and b represent the
lengths of the legs, and c represents the
length of the hypotenuse.
En un triángulo rectángulo , el lado opuesto
al ángulo recto se llama la hipotenusa. Los
otros dos lados se llaman catetos. En la
figura de abajo , a y b representan la
longitudes de los catetos, y c representa la
longitud de la hipotenusa.
Existe una relación especial
entre las longitudes de los
catetos y la longitud de la
hipotenusa. Esta relación se
conoce hoy como el Teorema
de Pitágoras
hipotenusa
catetos
There is a special relationship between the lengths of the legs and the length of the
hypotenuse. This relationship is known today as the Pythagorean Theorem.
JRLeon
Geometry Chapter 9.1
HGHS
Lesson 9.1
The Three Sides of a Right Triangle
c
a
JRLeon
b
Los tres lados de un triángulo rectángulo
c
a
b
c
a
b
Geometry Chapter 9.1
c
a
b
HGHS
Lesson 9.1
Los tres lados de un triángulo rectángulo
The Three Sides of a Right Triangle
Area of Large Square is: (c)(c)=c2
c
ba
c
b
a
a
c
b
c ab
Area del cuadrado grande
Length of Small Square = (b – a)
Area del cuadrado pequeño
So the base (base) = (b – a) and the height (altura) = (b – a)
This means that the Area of Small Square
Esto significa que el área de la
= (b – a)2
The Area of the 1 triangle =



The Area of the 4 triangles = 


 = 2
Area of the large square = Area of the small square PLUS the Area of the 4 Triangles
Área del cuadrado grande = área del cuadrado pequeño MÁS el área de los 4 triángulos
c2= (b – a)2 + 2
c2= b2 – 2ab + a2 + 2
c2= b2 – 2ab + a2 + 2
c2= a2 + b2
JRLeon
Teorema de Pitágoras
En todo triángulo rectángulo el cuadrado de la
hipotenusa es igual a la suma de los cuadrados de
los catetos.
Geometry Chapter 9.1
HGHS
Lesson 9.1
The Pythagorean Theorem works for right triangles, but does it work for all triangles?
A quick check demonstrates that it doesn’t hold for other triangles.
JRLeon
Geometry Chapter 9.1
HGHS
Lesson 9.1
JRLeon
Geometry Chapter 9.1
HGHS
Lesson 9.2
Three positive integers that work in the Pythagorean equation are called
Pythagorean triples.
El inverso del Teorema de Pitágoras
Si las longitudes de los tres lados de un triángulo satisfacen la
ecuación de Pitágoras, entonces el triángulo es un triángulo
rectángulo.
JRLeon
Geometry Chapter 9.2
HGHS
Lesson 9.3
In this lesson you will use the
to discover some relationships
between the sides of two special right triangles. One of these special triangles is an
isosceles right triangle, also called a 45°-45°-90° triangle. Each isosceles right triangle is half
a square, so these triangles show up often in mathematics and engineering.
En esta lección usted usará el
para descubrir algunas relaciones entre los lados de
dos triángulos rectángulos especiales . Uno de estos
triángulos especiales es un triángulo rectángulo
isósceles, también llamado un 45 °-45 °-90 ° triángulo.
Cada triángulo rectángulo isósceles es la mitad de un
cuadrado, por lo que estos triángulos aparecen a
menudo en las matemáticas y la ingeniería
JRLeon
Geometry Chapter 9.3
HGHS
Lesson 9.3
Investigation 1
In this investigation you will simplify radicals to discover a relationship between the
length of the legs and the length of the hypotenuse in a 45°-45°-90° triangle.
To simplify a square root means to write it as a multiple of a smaller radical without
using decimal approximations.
Length of each leg
1
2
3
4
5
6
7
...
10
...
l
Length of hypotenuse
JRLeon
Geometry Chapter 9.3
HGHS
Lesson 9.3
C
Given: Equilateral ABC
AC  CB  AB , Equilateral Triangle Definition
60°30°
Construct Perpendicular Angle Bisector CD.
2a
AD  BD , Perpendicular Angle Bisector
DCB = 30° , Perpendicular Angle Bisector
Let DB = a
A
JRLeon
60°
60°
D
a
B
Geometry Chapter 9.3
Then CB = 2a
HGHS
Lesson 9.3
Investigation 2
Another special right triangle is a 30°-60°-90° triangle, also called a 30°-60° right triangle,
that is formed by bisecting any angle of an equilateral triangle. The 30°-60°-90° triangle
also shows up often in mathematics and engineering because it is half of an
equilateral triangle. In this investigation you will simplify radicals to discover
a relationship between the lengths of the shorter
and longer legs in a 30°-60°-90° triangle.
Length of shorter leg
1
2
3
4
5
6
7
Length of hypotenuse
2
4
6
8
10
12
14
...
10
20
...
a
2a
Length of longer leg
JRLeon
Geometry Chapter 9.3
HQHS
Lesson 9.1 / 9.2
Class Work / Home Work:
9.1 Pages 481: problems 1 thru 18 EVEN
9.2 Pages 486-487 : problems 1 thru 18 EVEN
9.3 Pages 493: problems 1 thru 8 ALL
JRLeon
Geometry Chapter 9.1-9.2
HGHS

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