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parent_permlink | hive-196387 |
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author | meta007 |
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permlink | another-interesting-conclusion-from-a |
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title | "Another Interesting Conclusion from a Non-hypotenuse Side and In-radius of a Right Triangle" |
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body | "Hello math bugs(🐞) and hivers(🐝)
I hope everything with okay with you.
Well come to another geometric problem and it's solution.
We have a right angle triangle. If in-radius(**r**) of circle inside the triangle and a side (**a**=non hypotenuse) are given, we can find the area using both of them. Area in terms of the **a** and **r** can be given by : ∆= **ar**(**a**-**r**)/(**a**-2**r**)
![](https://images.ecency.com/DQmf8Qtf9kRaRvWXE2K872PYmhBuz4WxLMrYGu13ze6Zs3Q/addtext_07_26_05.19.38.png)
To prove the above consideration, what we need are follows:
☑️ Lentgh of a common tangents from a fixed point outside a circle are equal in lenght:
![](https://images.ecency.com/DQmb2bBgS9UAzchsTQRXjJXb1CMZL6YdWq52vP7jn43LyBs/addtext_07_28_08.59.57.png)
☑️☑️ The second thing we need here is the legendary Pythagoras theorem:
**H**ypotenuse² = **P**erpendicular² + **B**ase². [Details](https://ecency.com/hive-163521/@meta007/understanding-pythagorean-triplet)
**LET'S PROVE ∆ = **ar**(**a** - **r**)/( **a** - **2r**) :
As we know common tangents from outer point of a circle are equal in length, we can have three pairs of common tangents from three outer points (**A**, **B** & **C**) of **O**-centric in-circle with radius **r**. Check the following figure: 👇
![](https://images.ecency.com/DQmRyEfbgYP7ohi9i8c5TgGUMp1S4FSrEXF5yBii5Tw3yAh/addtext_07_28_10.00.53.png)
Now, from right ∆ABC above, we can say
**AB**² + **BC**² = **AC**²
**Or**, (**x** + **r**)² + **a**² = {**x** + (**a** - **r**)}²
**Or**, x² + r² + 2rx + a² = x² + (a - r)² + 2x(a -r)
**Or**, r² + 2rx + a² = (a - r)² + 2ax - 2rx
[ x² is removed from both side ]
**Or**, r² + a² + 4rx = a² + r² -2ar +2ax
**Or**, 4rx = 2ax - 2ar
[(a² + r²) is removed from both side]
**Or**, 2rx = ax - ar [ Dividing both side by 2]
**Or**, 2rx - ax = - ar
**Or**, ax - 2rx = ar
**Or**, x(a - 2r) = ar
**Or**, x = ar/(a - 2r)
LET'S GO TO THE FIGURE AGAIN
Height **AB** = (x + r) unit = r + ar/(a - 2r) unit
Base **BC** = a
Ar. ∆ABC = 1/2 × BC × AB unit²
**Or**, ∆= 1/2 × a × { r + ar/(a - 2r)} unit ²
Check the following figure now :
![](https://images.ecency.com/DQma8f3v8Spu7VKYfKkLGbz7VhiZAuKbgcFHjCby6b2yL4Y/addtext_07_28_11.01.50.jpg)
🎤🎤All the figures and drawing are made by me using android only. There may be some silly mistakes; excuse me for that and the figure may not be accurate ; please try considering the data only.
Thank you so much guys for stoping by
I hope you have liked it
Have a great day
All is well
Regards: @meta007" |
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