# Find the legs of isosceles triangle, given only the base

Is it possible to find the legs of isosceles triangle, given only the base length? I think that the info is insufficient. Am I right?

-
Yes, a sketch or two shows that. – André Nicolas Jun 5 '13 at 14:59

You are correct that it is impossible. Given only the base length of an isosceles triangle, we cannot determine the length of its sides: one would need to have the measure of the angle opposite the base in order to determine the lengths of the sides.

If the base of a triangle is fixed, an angle of smaller measure $m$ opposite the base would give longer congrent sides, than would an angle of greater measure. See for example, the following nested triangles:

For the same base $\overline{AC},\;m(\angle E) \lt m(\angle D) \lt m(\angle B)$, and $|CE|>|DE|> |BE|$.

You can experiment with a triangle of a given base, to see how the angle opposite the base determines the length of its sides.

-
Neat applet! +1 – Amzoti Jun 6 '13 at 0:40
@Amzoti I wish I could have posted it, here, in action! – amWhy Jun 6 '13 at 0:43
@Amzoti I just uploaded an image from geogebra: I finally figured out that you can export graphics directly to image files (e.g., png) – amWhy Jun 6 '13 at 1:18
@Amzoti Nice...I haven't explored the full power of WA, nor of my neglected Mathematica/Matlab software...(blame it on MSE - AGAIN) ;-) – amWhy Jun 6 '13 at 1:24
Wow, I don't have either of those, and want Mathematica so bad! It is an amazing piece of SW!!!! Matlab is used by a lot of engineers as it has wonderful communications add-ons. – Amzoti Jun 6 '13 at 1:25

The base length of an isosceles triangle is not enough to determine the triangle:

$\hspace{2cm}$

-

Given the base langth $a$ any $b>\frac a2$ constitutes a valid leg length.

-

You will always need three data to determine a triangle. It can be any combination of angles, lengths, heights...

In your example, since you know it's an isosceles triangle, you'll have three data once you define an angle or a length apart from the base's length, because you know both angles at the base's ends are equal.

So, if you have the segment AB being the base of an isosceles triangle, once you know either angle C, height from base to C, length BC or length AC, you'll have everything you need.

-
"It can be any combination of angles, lengths, heights..." wrong. How long are the sides on any given equilateral triangle? You have three pieces of data - 3 angles, each of 60°, but you don't have anywhere near enough information to determine the length of the sides. -1 (if I had to rep). – mikeTheLiar Jun 5 '13 at 19:55
@mikeTheLiar I don't think my answer was wrong completely... if you know the three angles of a triangle, you can never know the size, but you know the shape, which is sufficient to answer the question. Let's change it to: 'Given three data, you can find any triangle given you have one length among these data, otherwise you get a similar triangle with unknown size" – Óscar Gómez Alcañiz Jun 8 '13 at 17:00