In triangle abc
angle a=3x+40
angle b= 8x+35 and angle c=10x
Which is the longest side of the triangle, ab, ac or bc?
3x+40 + 8x+35 + 10x = 180
21x = 105
x = 5
a = 55
b = 75
c = 50
since angle b is the greatest, side ac is the longest.
To find the longest side of the triangle, we need to compare the lengths of the sides AB, AC, and BC.
The sum of angles in a triangle is always 180 degrees. Therefore, we can write the equation:
angle A + angle B + angle C = 180
(3x + 40) + (8x + 35) + (10x) = 180
Simplifying the equation:
3x + 8x + 10x + 40 + 35 = 180
21x + 75 = 180
Subtracting 75 from both sides:
21x = 105
Dividing by 21:
x = 5
Now that we know the value of x, we can substitute it into the expressions for the angles:
angle A = 3x + 40 = 3(5) + 40 = 55 degrees
angle B = 8x + 35 = 8(5) + 35 = 75 degrees
angle C = 10x = 10(5) = 50 degrees
Now let's calculate the lengths of the sides AB, AC, and BC using the Law of Sines:
AB/sin(angle A) = AC/sin(angle B) = BC/sin(angle C)
AB/sin(55) = AC/sin(75) = BC/sin(50)
Let's call the lengths of the sides AB, AC, and BC as a, b, and c respectively. We can simplify the equations as follows:
a/sin(55) = b/sin(75) = c/sin(50)
Now, we need to find which side has the largest value. To do that, we need to compare the ratios:
a/sin(55) < b/sin(75) < c/sin(50)
Since the sine values are positive, we can compare the lengths of the sides directly:
AB < AC < BC
Therefore, the longest side of the triangle is BC.
To determine which side is the longest in triangle ABC, we need to compare the lengths of the sides. However, the given information only provides the measures of the angles in terms of 'x'. Without the lengths of the sides, we cannot directly determine which side is the longest.
To find the longest side, we need additional information such as the lengths of the sides or the relationships between the sides and angles (e.g., the Law of Sines or Law of Cosines). Please provide any other relevant details or measurements to further analyze the triangle.