Hypotenuse known , the ratio width : height known, How to find width and height value?

Question

Premises:

The nature of the problem is with respect to TV dimension. I came across this when i was planing fo TV space required to mount onto my Wall.

Question:

How to find width and height of rectangle?

Known Value.

• the ratio of width to height is 16:9.
• the hypotenuse is 42 inch (106.68cm)

Answer 1

It's simpler to solve it formally: let $H$ be the hypotenuse, $h,w$ the height and width, $r$ the ratio $w/h$. Just apply Pythagoras' theorem: $$H^2=h^2+w^2= (1+r^2)h^2,\enspace\text{so }\quad h=\frac H{\sqrt{1+r^2}},\quad w=\frac {Hr}{\sqrt{1+r^2}}.$$

Answer 2

Write $w=16k$, $h=9k$ to get $42^2=w^2+h^2=16^2 k^2+9^2 k^2=(256+81)k^2=337 k^2$. Then $k=\frac{42}{\sqrt{337}}$ and therefore $w=\frac{42}{\sqrt{337}}16$ and $h=\frac{42}{\sqrt{337}}9$. Note that $\frac{42}{\sqrt{337}}\approx 2.28$.

Answer 3

You can do this with just trig functions

$$w = opposite$$ $$h = adjacent$$ $$r (ratio) = \frac{w}{h} $$ $$H = hypotenuse$$ $$\arctan(\frac{w}{h}) = A$$ $$\sin(A) = \frac{w}{H}$$ $$\cos(A) = \frac{h}{H}$$ $$h = \sin(\arctan(r)) * H$$ $$w = \cos(\arctan(r)) * H$$