A tripod has equal length legs = a.

It sits on an isosceles triangular base, sides b,b,c.

It has height = h.

If the base is changed to sides b,c,c, what is the new height (as an expression)?

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- Jul 11th 2010, 09:24 AM #1

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- Jul 11th 2010, 11:58 PM #2
It's not clear to me what sort of "expression" you want for the new height.

The original height h can be calculated in terms of a, b and c. I get the relation $\displaystyle h^2 = a^2 - \frac{b^4}{4b^2-c^2}$. If the sides of the base are changed from b,b,c to b,c,c, that is equivalent to exchanging b and c. So the new height k would be given by $\displaystyle k^2 = a^2 - \frac{c^4}{4c^2-b^2}$. You could then derive a (messy) expression for the ratio k/h if that is what you wanted.

- Jul 13th 2010, 04:32 AM #3

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Hello Wilmer,

Creating an expression is too messy for me but given a,b,c the triangle which determines the altitude h consists of the following line segments

A perpendicular bisector of base triangle bbc

B perpendicular bisector of tripod triangle aac

C a

Using the cosine rule the angle between a and A can be determined. If this angle is K h =asinK

bjh

- Jul 13th 2010, 04:43 AM #4

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- Jul 13th 2010, 06:59 AM #5
Another method is to say that the sphere of radius $\displaystyle a$, centred at the top of the tripod, contains all three vertices of the triangle and hence contains its whole circumcircle. If the radius of the circumcircle is $\displaystyle r$, you then have a right-angled triangle with sides $\displaystyle a$ (hypotenuse), $\displaystyle r$ and $\displaystyle h$. I used the formula for the circumradius, together with Pythagoras, to get the expression for $\displaystyle h$.