So there is a right triangle $ABC$ with $m∠C=90°$, $m∠B=75°$, and $BC\
(the \ hypotenuse)=12 cm$. I want to find the area of this triangle
It would look something like this:
Note: I have already solved this problem and got the answer as $18$ $cm^2$. So, I am not looking for an answer, I am looking for another way to solve this problem.
I have looked at other stack exchange questions similar to this question that involves $15-75-90$ triangles:
Ex. https://math.stackexchange.com/a/2082666/521593
All of these questions were solved using trigonometric functions however, I think there is a way to solve this using elementary geometry without trigonometric functions. I tried to go somewhere with splitting $∠B$ into $30-60-90$ triangles or a $15-15-150$ triangle but to no avail as it did not help me at all.
If anyone could help, find this way it would be most appreciated. Thanks.
Best Answer
Here is your triangle with just one extra segment inscribed in it:
Now you have a $30$-$60$-$90$ triangle, whose ratios you presumably know. That is, you know the ratios of $AC$ and $AD$ to $CD.$ But also $BD=CD$ and $AB = AD + BD,$ so you have the ratio $AB : CD,$ and now you can use the Pythagorean Theorem to get the ratio $BC: CD.$ But $BC = 12,$ and using the ratios you have found you can assign lengths to all the other segments, in particular $AB$ and $AC.$ Then you can find the area.