What is the length of the side opposite to a 30 degree angle if it is 12 units less than the hypotenuse of the triangle?
30-60-90 special triangle has sides in the ratio of
1 : sqrt 3 : 2 (a:b:c)
sin A = a/c = 1/2
hypotenuse = c = h
side a = h - 12 (opposite 30 deg angle)
set-up a ratio
1/2 = (h - 12)/h
cross multiply
h = 2(h - 12)
h = 2h - 24
-h = -24
h = 24
hypotenuse = 24
side a (opposite 30 d angle)= h - 12
side a = 24 - 12
side a = 12
length of side = 12 units
To find the length of the side opposite to a 30-degree angle in a triangle, we need to use trigonometric ratios. In this case, we can use the sine ratio.
The sine ratio states that the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right triangle is equal to the sine of that angle.
Let's assume that the length of the hypotenuse is represented by "h". Since the side opposite to the 30-degree angle is 12 units less than the hypotenuse, we can represent it as "h - 12".
Now, we can set up the equation using the sine ratio:
sin(30°) = (h - 12) / h
The sine of 30 degrees is equal to 0.5, so the equation becomes:
0.5 = (h - 12) / h
To solve for "h", we can cross-multiply:
0.5h = h - 12
Multiplying both sides by 2 to eliminate the decimal:
h = 2(h - 12)
h = 2h - 24
Subtracting "h" from both sides:
0 = h - 24
Adding 24 to both sides:
24 = h
Therefore, the length of the hypotenuse is 24 units.
To find the length of the side opposite to the 30-degree angle, we substitute the value of the hypotenuse back into the equation:
side opposite to 30° = h - 12
side opposite to 30° = 24 - 12
side opposite to 30° = 12 units