## Yahoo! 3 years ago In a right triangle , if the hypotenuse is four times as long as the perpendicular from the opposite vertex , one of the acute angles is?

1. jhonyy9

can you draw it please ?

2. Yahoo!

pic us not given..

3. jhonyy9

indifferent just for what is your oppinion from this pic

4. heena

|dw:1339328315880:dw| i gues its like dis

5. apoorvk

|dw:1339328315376:dw|

6. heena

my ans is theta=cos-1(1/4)

7. apoorvk

8. heena

i did and i dont find any prob as its saying hypotenuse is four times as long as the perpendicular means we cas asume perpendicular as x and hypotenuse as 4x

9. AndrewNJ

|dw:1339328062826:dw| i draw the pic u can find ab=4x^2 ---1 and a^2+b^2=16x^2 ----2 and use ab to represent x^2 at equation 2 we can have a^2+b^2=4ab devide ab two sides, that is a/b+b/a=4 we can let a/b=tanx (x could be the angle of one of two unknown angles) so tanx+1/(tanx)=4 let tanx=n so n+1/n=4, we can rewrite it as n^2-4n+1=0 and $n=2\pm \sqrt{3}$ $\tan x= 2\pm \sqrt{3}$ and u can work out the value of x that is the angle

10. Yahoo!

The options 60 45 30 15

11. FoolForMath

The correct answer is $$15^\circ$$

12. Yahoo!

@FoolForMath u r correct how????? PLzz show it!

13. apoorvk

lol sorry!! $tan(15^o) = 2 - \sqrt3$ and the other angle will automatically be 90-15 = 75 degrees.

14. FoolForMath

Well, I can think of a radical free beautiful solution :)

15. Yahoo!

i did nt understand the method used!

16. apoorvk

Don't just think, share it with us. @FoolForMath -_-

17. Yahoo!

@AndrewNJ can u xplain it!

18. AndrewNJ

in my answer, the equation 1 ab=4x^2 for area of the triangle equals 0.5ab or 0.5*4x*x

19. Yahoo!

No Idea!

20. AndrewNJ

the equation 2, a^2+b^2=4ab means Pythagorean Theorem

21. Yahoo!

Side a is hypot

22. AndrewNJ

ok, dont be serious about a and b which is longer, it doesnt matter a is perpendicular to b so 0.5ab is the area of this triangle

23. AndrewNJ

and half of hypotenuse times its attitude is also the area of triangle

24. AndrewNJ

so these two part, 0.5ab and 0.5*4x*x can be equal

25. AndrewNJ

this is the first equation

26. AndrewNJ

the second is accroading to Pythagorean Theorem can u got it?

27. AndrewNJ

by the way hypotenuse is the longest side in a right triangle

28. FoolForMath

|dw:1339329313013:dw| $\frac{AM}{MB }= \tan \alpha$ $\frac{AM}{MC}=\cot \alpha$ Thus,$MB+MC =AM(\cot \alpha +\tan \alpha)$ $\implies BC = 4AM=2AM \frac{1}{ \sin 2 \alpha}$ $\implies \sin 2\alpha =\frac 12 \implies \alpha = 15^\circ$

29. Yahoo!

@AndrewNJ i have a doubt in ur 1 st eq a^2=4x^2 is that right?

30. AndrewNJ

my first eq is 0.5ab=4x^2 !

31. AndrewNJ

@FoolForMath yours looks cool! but i think your method could be a bit difficult

32. FoolForMath

why? I think it's fairly obvious :)

33. AndrewNJ

well as far as I'm concerned, i seldom use cot sec :(

34. AndrewNJ

high school in my country didnt teach us cot sec,only taught sin cos and tan

35. FoolForMath

which country is that?

36. AndrewNJ

guess it!

37. FoolForMath

I am very bad at guessing :(

38. AndrewNJ

alright, it's china!

@AndrewNJ, I am from Missouri, the "show me" state lol, and I would like you to please show me how ab=4x^2 (equation 1 in your first post)

40. Yahoo!

i also want it @AndrewNJ

41. AndrewNJ

for 0.5ab is the area of triangle and 0.5*4x*x is also the area of triangle so 0.5ab=0.5*4x*x divide 0.5 in two sides is ab=4x^2

Thanks, you had already shown that, just that, I was being to dense to see it. Thanks again

*to too

44. AndrewNJ

:)