Question:
If π and π are two sides of a right angled hypotenuse π and let π be the perpendicular from the opposite
vertex on the hypotenuse, show that:
1. (πΏ π/πΏπ)bΒ =b3/c3
2. (πΏπ/πΏπ )A=b/c
1. From Pythagoras theorem
The area of the triangle
Then
"p=p(a, b)=\\dfrac{ab}{\\sqrt{a^2+b^2}}"
Differentiate by "a"
"=\\dfrac{a^2b+b^3-a^2b}{\\sqrt{a^2+b^2}(a^2+b^2)}"
Substitute "c=\\sqrt{a^2+b^2}"
2.
Let "a=kb, k>0." Then "c^2=a^2+b^2=k^2b^2+b^2=b^2(1+k^2)."
"p=\\dfrac{b}{c}(a)"
Use that "\\dfrac{b}{c}=const," if "A=const." Then
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