The angles of a triangle 📐 are in the ratio 3:4:2. If the shortest side is 5cm , calculate the length of the longest side?

Let the angles of the triangle be 3x, 4x, and 2x.

Since the sum of the angles in a triangle is 180 degrees, we have:

3x + 4x + 2x = 180
9x = 180
x = 20

Therefore, the angles are 60 degrees, 80 degrees, and 40 degrees.

Let the sides of the triangle be a, b, and c corresponding to the angles 60, 80, and 40 degrees respectively.

Using the sine rule, we have:

a/sin(60) = 5/sin(80) = c/sin(40)

a/sqrt(3)/2 = 5/sqrt(3)/2 = c/sqrt(3)/2
a = 5sqrt(3)
c = 5sqrt(3)/2 * sqrt(3)/2 = 15/2 = 7.5

Therefore, the length of the longest side is 7.5 cm.

Let the angles of the triangle be 3x, 4x and 2x, where x is a constant.

Since the sum of the angles of a triangle is 180 degrees, we have:

3x + 4x + 2x = 180
9x = 180
x = 20

Therefore, the angles of the triangle are:

3x = 3 - 20 = 60 degrees
4x = 4 - 20 = 80 degrees
2x = 2 - 20 = 40 degrees

The shortest side is given as 5 cm. Let the other two sides be y cm and z cm.

By the Law of Sines, we have:

y/sin 60 = z/sin 80 = 5/sin 40

Since the shortest side is 5 cm, y must be equal to 5.

Therefore,

z/sin 80 = 5/sin 40
z = 5 - sin 80 / sin 40
z = 5 - 0.9848 / 0.6428
z = 7.8 cm

Hence, the length of the longest side is 7.8 cm.