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I have two right angled triangles

The first one has an Opposite of $2.5$ & a Hypotenuse of $7.5$

The second has an Opposite of $10$ & a Hypotenuse of $45$ The ratio changes with the increase in the value of the opposite. Please can anyone derive a formula to find the hypotenues when the adjacent is, for example, 1 bearing in mind the changing ratio of triangles 1 & 2

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  • $\begingroup$ Which ratio changes? Does the length of the hypotenuse also change? $\endgroup$ – GoodDeeds Feb 29 '16 at 9:27
  • $\begingroup$ Not enough data to pin down the relations among the sides. $\endgroup$ – Gerry Myerson Feb 29 '16 at 9:35
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The ratio called $\sin$ in the two cases 1,2 is changing!

$$ \sin \alpha = \frac{2.5}{7.5}= \frac{1}{3} $$

$$ \sin \beta = \frac{10}{45} = \frac{1}{4.5} $$

Please note, the ratio changes if either of two, of opposite side or hypotenuse, changes.

The hypotenuse in case 1 is:

$$ \frac{1}{\cos \alpha}$$

and in case 2 is:

$$ \frac{1}{\cos \beta}.$$

EDIT1:

Hypotenuse length =

$$ \frac{adj. side}{\cos \alpha} \;\; OR \;\; \frac{adj. side}{\cos \beta} $$ OR $$ \frac{opp. side}{\sin \alpha} \;\; OR \;\; \frac{opp. side}{\sin \beta} $$

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  • $\begingroup$ So the formula for finding the hypotenuse is? $\endgroup$ – Banny Feb 29 '16 at 9:44
  • $\begingroup$ edited answer. Clear? $\endgroup$ – Narasimham Feb 29 '16 at 9:54
  • $\begingroup$ Much clearer, thank you Narasimham $\endgroup$ – Banny Feb 29 '16 at 10:10
  • $\begingroup$ However, where is the change in the ratio? In the case 1 the ratio of the length of the hypotenuse to the adjacent is 3:1 In case 2 it's 4.5:1 is what is the increment per unit change in the adjacent? So to repeat the original question, what would the length of the hypotenuse be if the adjacent was 1 using the decrement ratio of 4.5:1>3:1 $\endgroup$ – Banny Feb 29 '16 at 15:35
  • $\begingroup$ Sorry, confusion in terms. I MEANT: However, where is the change in the ratio? In the case 1 the ratio of the length of the hypotenuse to the OPPOSITE is 3:1 In case 2 it's 4.5:1 is what is the increment per unit change in the OPPOSITE ? So to repeat the original question, what would the length of the hypotenuse be if the OPPOSITE was 1 using the decrement ratio of 4.5:1>3:1 $\endgroup$ – Banny Feb 29 '16 at 15:45

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