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90 Degree Angle Triangle

In 90 degree angle triangle, two of the sides are sides of the right angle. These are called the legs of the right triangle. The third side is opposite the right angle. It is called the hypotenuse of the right triangle. Call the lengths of the legs a and b, call the length of the hypotenuse c. the Pythagorean Theorem says the a2+b2=c2. This fact only applies to right triangles.

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The Pythagorean Theorem 90 degree angle triangles:


This is due to finding right triangle in abundance in the physical structures of the world around us and the importance of finding perpendicular distance. The very important tools in 90 degree angle triangle problem are the Pythagorean Theorem


The Pythagorean Theorem:

 More unlike proofs have been written for the Pythagorean Theorem than for any other theorem in geometry. Pythagoras is given credit for developing the follow proof based on similar triangles.

Given right ∆ABC with the right angle at C, prove a2+b2=c2.


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Altitude `barCbarD` form two 90 degree angle triangles that are similar to each other and to the original triangle. Therefore,

`c/b`=`b/m`       or         `c/a`=`a/m`

So, b2=_(anyvalue)_ and a2=_(anyvalue)_, adding, a2+b2=c2


Example for 90 degree angle triangles:


Problem 1:

Derive a formula for the diagonal d of a square in terms of any side s using Pythagorean Theorem.

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Solution:

 c2=a2+a2

c2=2a2

Answer: c="a`sqrt(2)`

Problem 2:

Derive a formula for the altitude h of any equilateral triangle in terms of any side s using Pythagorean Theorem.

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Solution:

h="s-(`s/2`)

h="s-`s/4`=`(3s)/(4)`

Answer: h="`(s)/(2sqrt(3))`

Two basic properties of similar triangles using Pythagorean Theorem:

 Rule 1:

If two 90 degree angle triangles are similar using Pythagorean theorem

a)      Their corresponding angles are congruent.

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Thus, if ∆I~∆I’ in figure

Then

`angle`C’`~=``angle`C

`angle`A’`~=``angle`A

`angle`B’`~=``angle`B

And     m `angle`C=90o

m`angle`A=40o

m `angle`B=50o


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b)      The ratios of their corresponding sides are equal.

Thus, if ∆I ~∆I’ in figure

Then c="15" since `c/15`=`9/3`

And b="12" since `b/4`=`9/3`.

Rule 2:

Ratios of corresponding sides of 90 degree angle triangle are equal using Pythagorean Theorem

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If ∆II’~∆II in figure, find x and y by using the data indicated.

Solution:

Since ∆II’ ~ ∆II,

`(x)/(32)`=`(15)/(20)`                    `(y)/(26)`=`(15)/(20)` 

x="`15/20`32"                 y="`15/20`26

x="24"                      y="19`1/2`