Two sides of a triangle each have length of $5$. All of the following could be the length of the third side Except.
$(A)\quad 1$
$B \quad 3$
$C \quad 4$
$(D) \quad 7.07 \text{ or } \sqrt50$
$(E) \quad 10$
Do I use the formula $2 \sqrt{L^{2}}-A^2$ in order to find the base?
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$\begingroup$The answer is E , beacause then "the triangle will be straight line"(in the sense both the equal sides will lie on the base with the vertex angle i.e. the angle between the equal sides being $\pi$).
Using this type of idea you can also derive the celebrated trianle inequality which says,
"Sum of any 2 sides of a triangle is always grater than the third side".
$\endgroup$ $\begingroup$HINT: If you lay two $5$ inch rods end to end in a straight line, how long is that line?
(Note that since the question explicitly tells you that four of the five answers are possible lengths, it cannot be possible to determine the length of the base by some formula!)
$\endgroup$ $\begingroup$Use the triangle inequality theorem: Given sides of a triangle $a, b, c$ the sum of two of the sides must be greater then the remaining side in order for the triangle to exists. But you need a restriction for the exception,
SO $5+5=10$, thus any side below that would work.
$\endgroup$ $\begingroup$In order to solve this, you must know the vertex angle. I did some research and came up with a simple formula of base=pi(vertex angle)(2)(leg).
The answer is E, because of the formula a+b>c. In this problem, a+b=c, thus meaning that the triangle is not a triangle, but is a straight line.
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