I want to find the base of an isosceles triangle with base angles of 54.5 degrees and legs with the lengths of 43 with geometric relationships.
[Math] How to find the base of an isosceles triangle with the angles and legs
geometrytriangles
Related Solutions
The base length of an isosceles triangle is not enough to determine the triangle:
$\hspace{2cm}$
All you need to do is find one base angle, since in an isosceles triangle, both those angles are opposite equal sides, and are therefore equal.
Edit: Now that I see your image, You have enough information to find everything.
- You can use the Pythagorean Theorem to obtain the altitude (the unknown side in the middle.)
- The vertex, B, is split into two equal angles, but you can easily find the vertex angle by first finding one of the base angles.
- You can find the base angle B by using the fact that $$\cos B = \dfrac{\text{Adjacent}}{\text{Hypotenuse}}\implies B = \cos^{-1}\left(\dfrac{\text{Adjacent}}{\text{Hypotenuse}}\right) = \cos^{-1}\left(\frac{10.9}{25.6}\right) = 64.8$$
Each base angle, let's call them each of measure $x$, plus the vertex angle add up to $180$.
So if you need to solve for the vertex angle $\theta$, we have that $\theta = 180 - 2x$.
Also note, to find an angle, given none of the angles are known, you can to use the lengths of the sides of the triangle to solve for the base angles and/or vertex, and can use the Law of Cosines to do that.
Best Answer
Look at this picture http://mathworld.wolfram.com/images/eps-gif/Trigonometry_700.gif
Imagine that $\theta = 90-54.5= 35.5$.
This is a picture of your triangle cut in half, laid on its side, and shrunk to $1/43$ it's size.
We want to know what is the length of the side labeled "$\sin \theta $" is.
Your answer will be twice that multiplied by $43$.
So we plug it it into a calculator and get $\sin 35.5 = 0.58070295571093976105008.... $
So multiply that by $2$ and you will have the base of an isoceles triangle with legs of $1$.
Multiply that by $43$ and you will have the base of your tringle.
So the answer is $49.940454191140819450...... $.