Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I can't seem to find a textbook solution to this. It is always assumed that the length of the sides is know.

Isosceles triangle

Isolceles triangle

So the base $a$ is known. The bottom angles where $\alpha$ and the two sides $b$ touch, are known.

What is $h$?

share|improve this question
add comment

4 Answers

up vote 3 down vote accepted

enter image description here

solution sketch:

Note that $h$ divides the triangle into two right triangles (h is the perpendicular bisector (altitude) from the base to the opposite vertex); the angles line $h$ forms with $a$ are right angles. This gives you two right triangles, and you only need one of them to compute the values you need.

Call the two known (marked) angles $\theta$ (they are equal).

If you know the length of the side $b$: $\sin\theta = \dfrac{h}{b} \implies h = b\sin\theta$

If $a$ is known, $\tan\theta = \dfrac{h}{a/2} \implies h = \dfrac{a}{2}\tan \theta$.


Now, using the pythorean theorem to relate your sides, we know that

$$h^2+\left(\frac{a}{2}\right)^2\,=\;b^2\tag{pythogorean theorem}$$

If the length of $b$ is unknown, using the pythagorean theorem, then knowing $a/2$ and $h$ will allow you to solve for $b$.

Knowing $\,b\,$ and $\,h\,$ will allow you to solve for $\,\dfrac{a}{2}\,$ by the pythagorean theorem Then double the value of $a/2$ to get $a$.

share|improve this answer
    
Thank you, this seems to work. Likewise Inquisitive's answer. It's part of larger problem and i didn't get the results i expected, but that was a fault on my part. Your answer is correct. –  Adrian Jan 7 '13 at 15:17
    
You're welcome. Glad to help! –  amWhy Jan 7 '13 at 15:20
add comment

You can calculate h

1) using a and b

(a/2) * (a/2) + h * h = b * b

2) using sine law or cosine law http://www.transtutors.com/math-homework-help/laws-of-triangle/

h/ sinα = b / sin 90

share|improve this answer
add comment

from one of the following equations $$h^2+\left(\frac{a}{2}\right)^2=b^2$$ $$\sin\alpha=\frac{h}{b}$$ $$\cos\alpha=\frac{h/2}{b}=\frac{h}{2b}$$ $$\tan\alpha=\frac{h}{a/2}=\frac{2h}{a}$$ if you know two elements you can find the missing one

share|improve this answer
add comment

I have explained it graphically below.

Height

share|improve this answer
add comment

protected by Alexander Gruber 12 hours ago

Thank you for your interest in this question. Because it has attracted low-quality answers, posting an answer now requires 10 reputation on this site.

Would you like to answer one of these unanswered questions instead?

Not the answer you're looking for? Browse other questions tagged or ask your own question.