I am to find "d" (the hypotenuse)in the simplest radical form of a right triangle. The two angles are 30 degrees on top and the on the bottom is 60 degrees. Of course the opposite side to the left of 60 is 90 degrees. My options are:
A) 8sqrt3
B) 4
C) 8sqrt2
D) 16
I do not know how to figure this out
Since they did not give the length of
either side of the triangle, you'll
have to create a unit(1)triangle and
select d based on the INFO given.
let d=1, Y=1Sin60=0.8660=(sqrt3)/2=
side opp. 60 deg.angle, X=1Cos60=1/2=
side adj. to 60 deg. angle.The triangle is to be in simplest radical form. However, all sides must be multiplied by 4: d=4(1)=4, Y=4(sqrt3)/2
=2(sqrt3). So d=4 the smallest choice.
find ab in simplest form if A(0,3) and B(2,7)
To find the length of the hypotenuse, you can use the trigonometric function sine. In a right triangle, the sine function relates the ratio of the length of the opposite side to the length of the hypotenuse.
Since you know that the two angles in the triangle are 30 degrees and 60 degrees, you can determine the length of the hypotenuse by using the sine function.
The sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse:
sin(angle) = opposite / hypotenuse
In this case, the angle is 60 degrees, and the side opposite to it is d (the hypotenuse). So, we have:
sin(60°) = opposite / d
The sine of 60 degrees is √3/2. So, we can rewrite the equation as:
√3/2 = opposite / d
Since the opposite side is not given, we cannot directly solve for d. However, we can make use of the fact that in a 30-60-90 right triangle, the ratio of the side opposite the 30 degrees angle to the hypotenuse is always 1/2.
So, if we let the length of the side opposite the 30 degrees angle be x, we can set up the following equation:
1/2 = x / d
Now, we have two equations:
√3/2 = opposite / d
1/2 = x / d
To eliminate the variable d, we can divide both equations:
(√3/2) / (1/2) = (opposite / d) / (x / d)
√3 = opposite / x
Now, we can solve for the ratio of the opposite side to the side opposite the 30 degrees angle:
√3 = opposite / x
Rearranging the equation to solve for opposite:
opposite = √3 * x
Remember that we previously set the length of the side opposite the 30 degrees angle as x. So the opposite side can be written as:
opposite = √3 * x = √3 * (1/2)
Simplifying:
opposite = √3/2
From the given options, we can see that the only value that matches the length we found for the opposite side (√3/2) is option A) 8√3.
Therefore, the length of the hypotenuse in the simplest radical form of the right triangle is 8√3.