Question 1185773: The perimeters of two 30-60-90 triangles are in the ratio 1:2. If the hypotenuse of the larger triangle is 12, find the sides of the smaller triangle
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the triangles are in a 1:2 ratio for the corresponding sides and, consequently, for the perimeter.
the sides of the large triangle are 6 and 6 * sqrt(3), while the hypotenuse of the large triangle is equal to 12.
the sides of the small triangle are 3 and 3 * sqrt(3), while the hypotenuse of the small triangle is 6.
the corresponding sides are in a 1:2 ratio.
this means that the perimeter, which is the sum of all 3 sides, is also in a 1:2 ratio.
the area, which was not asked for, is in the ratio of 1^2 / 2^2 = 1:4.
the ara of the small triangle is equal to 1/2 * 3 * 3 * sqrt(3) = 4.5 * sqrt(3).
the area of the large triangle is equal to 1/2 * 6 * 6 * sqrt(3) = 18 * sqrt(3).
(18 * sqrt(3)) / (4.5 * sqrt(3)) is equal to 1/4.
the sides and the perimeter are in a 1:2 ratio.
the area is in a 1:4 ratio.
this all checks out.
your solution is that the sides of the smaller triangle are 3 and 3 * sqrt(3)
here's a reference.
https://www.onlinemath4all.com/trigonometric-ratio-table.htmlthe
the trig functions tell you the ratio of the sides to the hypotenuse (sine, cosine), the ratio of the hypotenuse to the sides (cosecant, secant), the ratio of one of the sides to the other (tangent, cotangent).
for example, in the 30-60-90 triangle, if the hypotenuse is 12, then:
the side opposite the 30 degree angle is equal to 1/2 * 12 = 6.
the side opposite the 60 degree angle is equal to sqrt(3)/2 * 12 = 6 * sqrt(3).
i won't get into tangent because that's not necessary for this problem.
if you need more, just write.
there are also lots of references on the web.
sometimes you need to dig to get the answer you're looking for.
you can usualy find what you need, but not always.
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