two angle of a triangle are 38 degree 27 minutes and 48 degree 42 minutes, respectively. if the longest side of the triangle is 256.6 cm ong find the length of the shortest side.
use the law of sines.
trigonometry
two angle of a triangle are 38 degree 27 minutes and 48 degree 42 minutes, respectively. if the longest side of the triangle is 256.6 cm ong find the length of the shortest side.
To find the length of the shortest side of the triangle, we need to use the law of sines, which states that the ratio of the length of a side of a triangle to the sine of the opposite angle remains constant for all sides and angles of the triangle.
First, we need to find the measure of the third angle. Since the sum of the interior angles of a triangle is always 180 degrees, we can subtract the given angles from 180 degrees to find the measure of the third angle:
Third angle = 180 degrees - (38 degrees 27 minutes + 48 degrees 42 minutes)
To subtract the angles with minutes, we need to convert the minutes to degrees. Since there are 60 minutes in a degree, we can divide the number of minutes by 60 to get the decimal value of the angle in degrees:
38 degrees 27 minutes = 38 + (27/60) = 38.45 degrees
48 degrees 42 minutes = 48 + (42/60) = 48.70 degrees
Third angle = 180 - (38.45 + 48.70) = 92.85 degrees
Now that we have the measures of all three angles, we can use the law of sines to find the length of the shortest side. The law of sines states that the ratio of the length of a side to the sine of its opposite angle is equal to the same ratio for any other side and its corresponding angle.
Let's name the longest side as side A (length = 256.6 cm), and the opposite angle as angle A. Similarly, name the shortest side as side C and the opposite angle as angle C.
Using the law of sines, we have:
side A / sin angle A = side C / sin angle C
Substituting the values we know:
256.6 cm / sin angle A = side C / sin 92.85 degrees
Now, we can solve for side C:
side C = (256.6 cm * sin 92.85 degrees) / sin angle A
To find sin angle A, we can use the angle sum property of triangles, which states that the sum of the angles in a triangle is always 180 degrees:
angle A = 180 degrees - (38.45 degrees + 48.70 degrees) = 92.85 degrees
Now, we can substitute the values into the equation:
side C = (256.6 cm * sin 92.85 degrees) / sin 92.85 degrees
Calculating this expression will give us the length of the shortest side of the triangle.