In this lesson you’ll learn about the Pythagorean Theorem, and related important properties that help to apply the theorem and its corollaries to solve real life problems. The presentation covering such content will be done by the instructor in own handwriting, using video and with the help of several examples with solution. This will help you understand important geometric relationships to solve problems from day to day life situations based on the above concepts, and how to make use of them.
From Wikipedia
You may review the earlier learning bout the triangles and their properties, especially for right triangle. In a right triangle, the area of the square whose side is the
hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
The theorem can be written in an equation form: a^{2} + b^{2} = c^{2}, where c represents the length of the hypotenuse,
and a, b represent the lengths of the other two sides.
(More text below video...)
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(Continued from above)In formulae
As stated above, if c be the length of the hypotenuse and a and b are the lengths of the other two sides, the theorem can be expressed as the equation: a^{2} + b^{2} = c^{2}. When solved for c and
d c =
(a^{2} + b^{2}) If c is given, the length of one of the legs can be found using the following equations:
c^{2} – a^{2} = b^{2} OR c^{2} – b^{2} = a^{2}
Thus if the lengths of any two sides are known, the length of the third side can be found. In general this theorem points towards the law of cosines;, using which given the lengths of two sides and the size of the angle between them, the length of the third side of a triangle can be determined. If the angle between the sides is a right angle, it reduces to the Pythagorean Theorem.
You can explore the Proofs of this theorem using similar triangles,
Euclid's or Garfield's proof, proof by subtraction, similarity proof, proof by rearrangement, Algebraic proof etc.
Pythagorean Theorem:
If a triangle is a right triangle, then the square of the length of the longest side is equal to the sum of the squares of the lengths of the other two sides (the legs). Refer Figure below:
Given ABC
is a right triangle with ACB
the right angle, c is the length of the hypotenuse, a and b are the lengths of the legs.
Prove c^{2} = a^{2} + b^{2}
ProofABC is a right triangle with ACB
= 90
Draw CDAB Now, let BD = x and AD = c  x.
Notice that the converse of the theorem is also true: it states “ If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle” .
A corollary of the above converse is a simpler way of determining whether a triangle is right, obtuse, or acute. Say c is chosen to be the longest of the three sides, remember as follows:
• If a^{2} + b^{2} = c^{2}, then the triangle is right.
• If a^{2} + b^{2} > c^{2}, then the triangle is acute.
• If a^{2} + b^{2} < c^{2}, then the triangle is obtuse.
According to another Theorem: “ A triangle is a right triangle if and only if the square of the length of the longest side is equal to the sum of the squares of the lengths of the other two sides”.
Pythagorean triples
A Pythagorean triple has three positive integers a, b, and c,
such that a^{2} + b^{2} = c^{2}.
In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. E.g. these are (3, 4, 5) and (5, 12, 13) etc., as 3^{2} + 4^{2} = 5^{2}.
The 4545Degree Right Triangle
If two triangles are isosceles right triangles then they are similar by AA~. An isosceles right triangle is called a 4545degree right triangle.
The 3060Degree Right Triangle
Each of the congruent right triangles formed by drawing an altitude to a side of an equilateral triangle is called a 3060degree right triangle. If two triangles are 3060degree right triangles, then they are similar by AA~.
Look at some of the examples explaining the application of Pythagorean Theorem to help you solve the Geometry problems.
Example 1: Given find out in the Figure below, missing length of the side in relation to other given dimensions.
Step 1: Use the formula:
Step 2: It may be rewritten as, c^{2} = a^{2} + b^{2}.
Step 3: Plug in the values of c and b i.e. c = 8.5, and b = 4.2,
it will be 8.5^{2} = a^{2} + 4.2^{2}.
Step 4: a^{2} = 8.5^{2}  4.2^{2}. On simplification
a = 7.4 cm.
Thus the missing length of side is 7.4 cm, as the final answer.
Example 2: Given Amy normally swims diagonally across the rectangular pool which is 11.6m (Figure below, NTS). However, on Sunday the pool was busy so she had to swim the length of the pool. The width of the pool is 5.2m. Find out how much short she did on distance in each swim?
When Amy swims diagonally across the rectangular pool, she covers 10.73m. When she had to swim the length of the pool, say she covers ‘b’ distance. First determine how much is ‘b’ in meters.
Step 1: Use the formula: c^{2} = a^{2} + b^{2}.
Step 2: Plug in the values of c and a i.e. c = 10.73 and a = 4.8m.
Step 3: The equation is in the form;
10.73^{2} = b^{2} + 4.8^{2}. b^{2} = 10.73^{2}  4.8^{2}.
Step 4: On simplification, it gives b = 9.6m
Thus the shortfall in the distance is (10.73  b)
= (10.73  9.6) = 1.13m, as the final answer.
The video above will explain more in detail about Pythagorean theorem,
with the help of several examples.
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