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Lesson 9: Areas of 2-D shapes (I)

May 7, 2011

Vocabulary-on-Plane Geometry 

Lesson 9: Areas (Notes)

Area of a Composite Figure and Word problems I

Area Worksheets II

I Have to Know by the End of this Lesson  

1. Pythagorean Theorem

A right-angled triangle is a triangle that has a right angle. We name hypotenuse to the side opposite the right angle (c on the picture).

 In a right-angled triangle the sum of the squares of the lengths of the sides equals the square of the length of the hypotenuse.

Notice that:

  • If we know the lengths of a side and hypotenuse, we can find the length of the other side
  • And if we know the lengths of the two sides we can find the length of the hypotenuse

When the side lengths of right triangle satisfy the Pythagorean Theorem, these three numbers are known as “Pythagorean triples or triplets”

How to identify and create Pythagorean triples?

The most common examples of Pythagorean triples are:

  •  3,4,5 triangles
  • A 3,4,5 triple simply stands for a triangle that has a side of length  3, a side of length 4 and a side of length 5.
  • If a triangle has these side length then it must be a right triangle
  • 5,12,13 right triangles
  • 7,24,25 right triangles
  •  8,15,17 right triangles

 We can conclude the following:

  • Only right triangles verify the Pythagorean Theorem
  • If a triangle has a triple as side lengths then it must be a right triangle

 Any multiple of the ratios above represent the sides of a right triangle.

 Here you are two interesting pages:

 On this page you can check that:

  • if you have that if you have an right-angled triangle:  

a2 + b2 = c2

  • if you have that if you have an acute-angled triangle:  

a2 + b2 < c2

  • if you have that if you have an obtuse-angled triangle:

a2 + b2 > c2

If you click on the picture  below you will find an animation the geometric demonstration of the Pythagorean Theorem (go down the page because the animation is at the end)

Finally this is a video on Pythagorean Teorem and its Converse (Is this triangle a right triangle?)

 2. Applications of the Pythagorean Theorem

 There are many applications of this famous theorem. One of them are the following:

  • Identifying right triangles
  • Calculating the diagonal of a rectangle
  • Calculating the height of an isosceles  triangle
  • Calculating the apothem of a regular polygon

If we remember the first I recommended, we can write the conditions that led us to classify the triangles according their angles:

  • a2 + b2 = c2 , then it is a right-angled triangle
  • a2 + b2 < c2 , then it is an acute-angled triangle
  • a2 + b2 > c2 , then it is an obtuse-angled triangle

On the following links you can practice exercises on Pythagorean Theorem.

Pythagorean theorem: find the length of the hypotenuse report 1 Pythagorean theorem: find the length of the hypotenuse

Pythagorean theorem: find the missing leg length report 2 Pythagorean theorem: find the missing leg length

Pythagorem theorem: find the perimeter report 3 Pythagorem theorem: find the perimeter

Pythagorean theorem: word problems report 4 Pythagorean theorem: word problems

Converse of the Pythagorean theorem: is it a right triangle? report 5 Converse of the Pythagorean theorem: is it a right triangle?

This is  a video where is solved a problem by using this theorem

 

 

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