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.

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

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

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Lesson 9.1

The Three Sides of a Right Triangle Los tres lados de un triángulo rectángulo

c a b c a b c a b c a b

JRLeon Geometry Chapter 9.1

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Lesson 9.1

The Three Sides of a Right Triangle Los tres lados de un triángulo rectángulo

Area of Large Square is: (c)(c)=c 2

Area del cuadrado grande

c a b c b a a c b b a c

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 =

𝟏 𝟐 𝒂𝒃 𝟒 𝟏 𝟐 𝒂𝒃

= 2

𝒂𝒃

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

c 2 = (b – a) 2 + 2𝐚𝐛 c 2 = b 2 – 2ab + a 2 + 2𝐚𝐛 c 2 = b 2 – 2ab + a 2 + 2𝐚𝐛 c 2 = a 2 + b 2

Teorema de Pitágoras

En todo triángulo rectángulo hipotenusa el cuadrado de la es igual a la suma de los cuadrados de los catetos .

JRLeon Geometry Chapter 9.1

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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

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Lesson 9.1

JRLeon Geometry Chapter 9.1

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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

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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

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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 Length of hypotenuse 1 2 3 4 5 6 7 ...

10 ...

l

JRLeon Geometry Chapter 9.3

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Lesson 9.3

A

60°

C D

60°30°

a 2a

60°

Given: Equilateral  ABC AC  CB  AB , Equilateral Triangle Definition Construct Perpendicular Angle Bisector CD.

AD  BD , Perpendicular Angle Bisector  DCB = 30° , Perpendicular Angle Bisector Let DB = a Then CB = 2a B

JRLeon Geometry Chapter 9.3

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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 Length of hypotenuse Length of longer leg 1 2 2 4 3 6 4 8 5 6 7 10 12 14 ...

10 20 ...

a 2a

JRLeon Geometry Chapter 9.3

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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

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