# Hypotenuse increases non linearly wrt the adjacent

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

• Which ratio changes? Does the length of the hypotenuse also change? – GoodDeeds Feb 29 '16 at 9:27
• Not enough data to pin down the relations among the sides. – Gerry Myerson Feb 29 '16 at 9:35

## 1 Answer

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

• So the formula for finding the hypotenuse is? – Banny Feb 29 '16 at 9:44
• edited answer. Clear? – Narasimham Feb 29 '16 at 9:54
• Much clearer, thank you Narasimham – Banny Feb 29 '16 at 10:10
• 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 – Banny Feb 29 '16 at 15:35
• 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 – Banny Feb 29 '16 at 15:45