Very dificult: The minimum perimeter and maximum height of a triangle under constraints

[loquetedigo]

Obtain
-The maximum height corresponding to the side b of any triangle (abc) once known the value of its perimeter and height corresponding to the a side a.
-The minimum perimeter of any triangle (abc) once known the heights corresponding to the a and b sides.
Aux:
Geogebra construction: http://tube.geogebra.org/student/m1176787
TrianCal (Triangle Calculator): TrianCal

 

[jvicens]

by 3n+1

Hello,

To answer your questions, I would like to introduce you to two helpful tools: Geogebra and TrianCal.

Geogebra is a free online tool that allows you to construct and manipulate geometric shapes. Using Geogebra, we can easily find the maximum height corresponding to the side b of any triangle (abc) once we know the perimeter and height corresponding to side a. I have created a Geogebra construction for you to see this visually. You can access it through this link: http://tube.geogebra.org/student/m1176787

As you can see in the construction, the maximum height of side b occurs when the triangle is isosceles, with sides a and b being equal. This creates a right angle at point C, making the height corresponding to side b the same as the height corresponding to side a. So, the maximum height of side b is simply equal to the height corresponding to side a.

Moving on to the minimum perimeter of any triangle (abc) once we know the heights corresponding to sides a and b, we can use TrianCal. TrianCal is an online triangle calculator that can solve for various properties of a triangle given certain information. You can access it through this link: TrianCal by 3n+1

Using TrianCal, we can input the known values of the heights corresponding to sides a and b, and it will give us the minimum perimeter of the triangle. This occurs when the triangle is equilateral, with all sides being equal. So, the minimum perimeter of any triangle (abc) once we know the heights corresponding to sides a and b is simply 3 times the height corresponding to side a (or b).

I hope this helps answer your questions. Thank you for your interest in geometry and using these helpful tools. Let me know if you have any further questions.