Each triangle is a 30-60-90 triangle, and the hypotenuse of one triangle is the longer leg of an adjacent triangle. The hypotenuse of the largest triangle is 8 centimeters. What is the number of centimeters in the length of the longer leg of the smallest triangle? Express your answer as a common fraction.
the answer is 9/2
There are 4 triangles
All four right triangles are 30-60-90 triangles. Therefore, the length of the shorter leg in each triangle is half the hypotenuse, and the length of the longer leg is sqrt3 times the length of the shorter leg. We apply these facts to each triangle, starting with triangle AOB and working clockwise.
From $\triangle AOB$, we find $AB = AO/2 = 4$ and $BO = AB\sqrt{3}=4\sqrt{3}$.
From $\triangle BOC$, we find $BC = BO/2 =2\sqrt{3}$ and $CO = BC\sqrt{3} =2\sqrt{3}\cdot\sqrt{3} = 6$.
From $\triangle COD$, we find $CD = CO/2 = 3$ and $DO = CD\sqrt{3} = 3\sqrt{3}$.
From $\triangle DOE$, we find $DE = DO/2 = 3\sqrt{3}/2$ and $EO =DE\sqrt{3} = (3\sqrt{3}/2)\cdot \sqrt{3} = (3\sqrt{3}\cdot \sqrt{3})/2 = \boxed{9/2}$
To solve this problem, we need to understand the properties of a 30-60-90 triangle.
In a 30-60-90 triangle, the ratio of the lengths of the sides is always 1:√3:2. This means that the longer leg is √3 times the length of the shorter leg, and the hypotenuse is twice the length of the shorter leg.
Let's label the sides of the triangles with variables to help us solve the problem.
Let x be the length of the shorter leg of the smallest triangle.
Since the hypotenuse of the largest triangle is 8 centimeters and is the longer leg of the adjacent triangle, we can set up the following equation:
2x = 8
To solve for x, we can divide both sides of the equation by 2:
x = 8/2
Simplifying, we get:
x = 4
Therefore, the length of the longer leg of the smallest triangle is 4 centimeters.
You should know the ratio of sides of your triangle to be:
1 : √3 : 2 or
x : √3x : 2x, where x is a constant
so in the next one:
a : 2x : c
then c/2x = 2x/√3x
√3 cx = 4x^2
c = 4x/√3
a/x = 2x/√3x
a = 2x/√3
general triangle: (2x/√3) : 2x : (4x/√3)
check: is a^2 + b^2 = c^2
LS = 4x^2/3 + 4x^2 = 16x^2/3
RS = (14x/√3)^2 = 16x^2/3
YES
so if 4x/√3 = 8
x = 2√3
our largest triangle is 4 : 4√3 : 8
our hypotenuse increased from 2 to 8, that is by a factor of 4, so to get to our smallest triangle largest leg we have to divide 4√3 by 4 to get √3
(which of course we knew at the beginning)
Draw the triangles. Recall that the sides are in the ratio 1:√3:2
Label the longest hypotenuse, and then work down to get the desired leg's size.
I get 6.
I am assuming there are only two triangles.