if a triangle has a perimeter of 72 and a ratio of 3:4:5. what is the length of all three sides and angles
3 x + 4 x + 5 x = 72
12 x = 72
x = 6
so
18
24
30
3,4, 5 is a right triangle
for the other angles
sin A = 3/5
sin B =4/5
To find the lengths of the sides of the triangle, let's assume the common ratio as 'x'.
According to the given ratio of 3:4:5, the three sides can be expressed as 3x, 4x, and 5x.
Since the perimeter of the triangle is given as 72, the sum of all three sides should be equal to 72. So, we can write the equation:
3x + 4x + 5x = 72
Combining similar terms, we get:
12x = 72
Dividing both sides by 12, we find:
x = 6
Now, substituting the value of 'x' back into the sides, we get:
Side 1 = 3x = 3 * 6 = 18
Side 2 = 4x = 4 * 6 = 24
Side 3 = 5x = 5 * 6 = 30
So, the lengths of the three sides of the triangle are 18, 24, and 30, respectively.
To find the angles, we can use the Law of Cosines, which states:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we can consider side 'c' as the longest side (30 in length), and the corresponding angle as angle 'C'. Sides 'a' and 'b' are the other two sides.
Using the values from above:
a = 18
b = 24
c = 30
Now, plugging these into the equation, we get:
30^2 = 18^2 + 24^2 - 2 * 18 * 24 * cos(C)
900 = 324 + 576 - 864 * cos(C)
900 = 900 - 864 * cos(C)
864 * cos(C) = 0
cos(C) = 0
To find the value of angle C, we know that cos(90 degrees) = 0. So, angle C is a right angle (90 degrees).
Since we have a right-angled triangle, the other two angles must add up to 90 degrees as well.
Let's call the other two angles A and B:
A + B + C = 180 degrees
By substituting the known values:
A + B + 90 = 180
A + B = 180 - 90
A + B = 90
Therefore, angles A and B are complementary angles of 90 degrees, but their specific values cannot be determined without more information.
In summary:
The lengths of the three sides of the triangle are 18, 24, and 30, respectively. Angle C is a right angle (90 degrees), and angles A and B are complementary angles. Their specific values cannot be determined without additional information.