# Math

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.

1. 👍
2. 👎
3. 👁
1. 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.

1. 👍
2. 👎
2. 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)

1. 👍
2. 👎
3. There are 4 triangles

1. 👍
2. 👎
4. 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 $EO =DE\sqrt{3} = (3\sqrt{3}/2)\cdot \sqrt{3} = (3\sqrt{3}\cdot \sqrt{3})/2 = \boxed{9/2}$

1. 👍
2. 👎

1. 👍
2. 👎

## Similar Questions

1. ### geometry

Find the value of x. Round the length to the nearest tenth. Diagram is not drawn to scale. A right triangle is drawn with right angle between a right vertical leg and lower horizontal leg. A dashed line is drawn horizontally from

2. ### Geometry

Find the value of x. Round to the nearest tenth of a unit. A right triangle. The hypotenuse is 580 yards. The opposite leg (not the base) is x. The exterior degree on the top of the triangle is 27. 263.3 yd 295.5 yd 516.8 yd

3. ### math

The hypotenuse of a right triangle measures 10m. One leg of the triangle is 2m longer then the other. Find the lengths of the legs. How would you solve this equation?

4. ### Algebra

Which statement about a right triangle is NOT true? A.) The hypotenuse is the longest side of the triangle. B.) The square of the hypotenuse is equal to the sum of the squares of the other two lengths. C.) The sum of the length of

1. ### Geometry

Which statement is true? A. The altitude drawn to the leg of a right triangle forms two triangles that are similar to each other and to the given triangle. B. The altitude drawn to the hypotenuse of a right triangle forms two

2. ### Math

1) The hypotenuse of an isosceles right triangle is 8 cm longer than either of its legs. Note that an isosceles right triangle is a right triangle whose legs are the same length. Find the exact length of its legs and its

3. ### math

I have 14 questions and I was wondering if someone could check my answers. This is a do or die situation. If I don't get an A, I may not get to graduate. I really appreciate you taking your time to double check theses for me.

4. ### Trigonometry

The hypotenuse of a right triangle is 34cm. find the length of the two legs, if one leg is 14cm longer than the other.

1. ### math

The shortest leg of a triangle is 2 inches shorter than the other leg. The hypotenuse of this triangle is 10 inches. What are the lengths of the two legs of this triangle?

2. ### Geometry/Algebra

The segments formed by the altitude to the hypotenuse of a right triangle are 8 and 10. Find the shorter leg of that triangle. Help please! I am confused on which formula to use or if I should use the Pythagorean Theorem.

3. ### algebra

The shorter leg of a right triangle is 8 ft shorter than the longer leg. The hypotenuse is 8 ft longer than the longer leg. Find the side lengths of the triangle. Length of the shorter leg: Length of the longer leg: Length of the