MHB Area of a Triangle from 3 sides

[Carla1985]

View attachment 6257

Can I have an opinion on this question, please? Personally I would use the cosine & sine rules to work out the angles then use trig to calculate the height. However, the question asks for Pythag to be used. Can someone please explain what method I should be using to answer this? Thanks

Thanks
Carla

 

[Euge]

Draw an altitude from the top vertex to the base who length is 21. The altitude divdes the triangle into two right triangles satisfying special Pythagorean triples. Which Pythagorean triples do these triangles satisfy?

 

[Carla1985]

Draw an altitude from the top vertex to the base who length is 21. The altitude divdes the triangle into two right triangles satisfying special Pythagorean triples. Which Pythagorean triples do these triangles satisfy?

Ok, I think I've got it.

So I have 2 right angle triangles with triples $(17,h,x)$ and $(10, 21-x,h)$. From pythag I get:

$$h^2+x^2=289$$
$$h^2+x^2-42x=-341$$

Subbing in $h^2$ into the second equation gives me $x=15$ and $h=8$. This gives me an area of $84$.

Thanks for the help!