Find the length of x and y in the 30-60-90" triangle. Leave answer in simplest radical form
Right triangle y, x, 6
X=? Y=?
Answer x= 3sqrt3 y= 3
In a 30-60-90 triangle, the sides opposite the angles measure x, 2x, and x√3 (where x is the shortest side).
Here, we know that the shortest side (opposite the 30 degree angle) is y, which means:
y = x
The side opposite the 60 degree angle is x√3, which means:
x√3 = 6
Solving for x:
x = 2√3
And solving for y:
y = x
y = 2√3
simplifying:
x = 2√3 = 3√3/2
y = 2√3 = 3√3/2
So, in simplest radical form, x = 3√3/2 and y = 3√3/2 (or we can express both as 3√3).
In a 30-60-90 triangle, the sides follow a specific ratio. The ratio is 1 : √3 : 2.
Given that y = 6, we can find x by multiplying y by √3.
x = y * √3 = 6 * √3 = 6√3
So, x = 6√3.
Therefore, x = 6√3 and y = 6.
To find the length of x and y in a 30-60-90 triangle, we can use the ratios of the sides.
In a 30-60-90 triangle, the ratio of the side lengths is:
x : y : 2y
Here, x is the shorter leg, y is the longer leg, and 2y is the hypotenuse.
Given that the longer leg, y, is 6, we can substitute this value into the ratio:
x : 6 : 2(6)
Simplifying the ratio, we have:
x : 6 : 12
To find the length of x, we can set up a proportion:
x / 6 = 6 / 12
Cross-multiplying, we get:
12x = 36
Dividing both sides by 12, we find:
x = 3
So the length of x is 3.
To find the length of y, we can substitute the value of x into the ratio:
3 : y : 12
Since we know that the side opposite the 60-degree angle is sqrt(3) times the shorter leg, we can conclude that y is:
y = 3 * sqrt(3)
So the length of y is 3sqrt(3).
Therefore, the final answer is x = 3sqrt(3) and y = 3.