triangle question?

Question

a right triangle has a 60 degree angle & a 30 degree angle. The hypotenuse is not given, but the other side is 9 in length.
i need to find the length of the base.
how would this be done?

Answer 1

180 - (60 + 30) = 180 - 90 = 90

So this is a right triangle.

tan(60) = 9/y
y = 9/tan(60)
y = 9/(sqrt(3))
y = (9sqrt(3))/3
y = 3sqrt(3)

x^2 + y^2 = c^2
9^2 + (3sqrt(3))^2 = c^2
81 + (sqrt(27))^2 = c^2
81 + 27 = c^2
c^2 = 108
c = sqrt(108)
c = 6sqrt(3)

ANS : Hypothenuse = 6sqrt(3), Base = 3sqrt(3)

Answer 2

There are several ways to solve it. Anyway, there are 2 solutions for this problem because the side 9 can be oppsite to any angle

You can use Sin law: sina / A = sinb / B (where A, B is the opposite angles of a, b respectively)
*if the 9 side is opposite to 30 degree angle
sin 60 / x = sin 30 / 9
sqrt(3) / 2x = 1/ (2*9)
sqrt (3) / x = 1 / 9
x = 9 * sqrt (3)
* if the 9 side is opposite to 60 degree angle
sin 60 / 9 = sin 30 / 9
solve the equation, we have x = 3 * sqrt3

You can also you Tangen = opp / leg
*if the 9 side is opposite the 30degree angle:
tan 30 = 9/x = 1/sqrt3
x = 9 *sqrt (3)

* if the 9 side is opposite the 60degree angle
tan 60 = 9/x = sqrt3
x = 9 / sqrt3 = 3 * sqrt3


So, there are 2 solution: 9*sqrt3 and 3*sqrt3, depending on where the 9 side is

Answer 3

It's easier than trig:

because there are two angles that are 30 and 60, you know it is a special triangle: the 30-60-90 triangle.

for all of these types of triangles, the base, height, and hypotenuse are proportional to 1, square root of three, and 2 respectively.

Therefore 9/x, if x is base, is equal to sqrt(3)/1.

then, just cross multiply, so sqrt(3)times x equals 9, then you can solve from there.

Answer 4

Trigonometric law :

For a 30-60-90 degrees triangles, the opposite side of 30 degres
angle is 1/2 of hypotenuse.

Triangle ABC

side a = c/2 = 9 => c=18

side a = 9

side b = ?


a^2 + b^2 = c^2

b^2 = c^2 - a^2

b= sqrt( c^2 - a^2)

= sqrt (18^2 - 9^2)

= sqrt 243

= 15,588

= 15.6

Answer 5

What do you mean by the "other side". There are 2, 1 opposite the 30° angle & the other opposite the 60° angle
they equal h/2 & h/2 √3 respectively.

if you mean the side opposite the smaller angle
h/2=9
h=18 in

if you mean the side opposity angle
h/2 √3=9
h√3=18
h=18/√3=18√3/3=6√3=10.39 in

so depending on what you mean by the other side, the answer is either
18 in or 10.39 in

Answer 6

is the 9 opposite the 30 or 60 degree angle?

trig is always useful

Answer 7

according to your question the 9 opposite to angle 60 (if U mean that)
let side opposite 30 be L then the hypotenuse is 2L
by using Pythagoras's theorem
(2L)^2-L^2=9^2
4L^2-L^2=81
3L^2=81 /3
L^2=27 then L=sq. root 27 =5.2
the length of the base =5.2

Answer 8

In a 30-60-90 triangle, the hypotenuse is always twice as long as the short leg. And the long leg is √3 times the short leg.

Can you finish the problem now?

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I think this problem belongs to the topic of special triangles in Geometry. Therefore, you don't need to use trigonometry.

Answer 9

This is a 90/60/30 angle. There is something you do with it. One is Square root of three times the smallest doubled...and the other...ah I can't remember.

Answer 10

With SOHCAHTOA.

Keep in mind that whatever rule applies,
sin(60) = sqrt(3)/2
cos(60) = 1/2
tan(60) = sqrt(3)
sin(30) = 1/2
cos(30) = sqrt(3)/2
tan(30) = 1/sqrt(3)

Which ever angle you use,

sinx = opposite/hypotenuse
cosx = adjacent/hypotenuse
tanx = opposite/adjacent