Q6. In a triangle if 4LA=3LB=12LC, find all the angles.
To find all the angles in a triangle when given the ratios of the lengths of its sides, you can use the Law of Cosines and the Law of Sines.
Let's assume that the measures of the angles in the triangle are A, B, and C, and the lengths of the sides are a, b, and c, respectively. According to the given information, we have:
4LA = 3LB = 12LC
This means that the length of side a is four times the length of side C, the length of side b is three times the length of side C, and the lengths of sides a, b, and c are in the ratio of 4:3:12.
To proceed, we need to assign a value to one of the lengths. Let's suppose that LC = 1 (you can choose any value as long as it satisfies the given ratios, and the angles will be the same).
Using the ratios, we can determine the lengths of sides a and b:
a = 4LC = 4(1) = 4
b = 3LC = 3(1) = 3
The value of c can also be found as the remaining side length:
c = 12LC = 12(1) = 12
Now, we can use the Law of Cosines to find one of the angles. The Law of Cosines states that for a triangle with sides a, b, and c, and angle A opposite side a, we have:
a^2 = b^2 + c^2 - 2bc * cos(A)
Substituting the given values:
4^2 = 3^2 + 12^2 - 2(3)(12) * cos(A)
16 = 9 + 144 - 72 * cos(A)
16 = 153 - 72 * cos(A)
72 * cos(A) = 137
cos(A) = 137 / 72
Using inverse cosine (cos⁻¹) or a calculator, we can find the value of A.
Similarly, we can apply the Law of Sines to find the other two angles. The Law of Sines states that for a triangle with sides a, b, and c, and angles A, B, and C opposite sides a, b, and c, respectively:
a / sin(A) = b / sin(B) = c / sin(C)
Substituting the given values:
4 / sin(A) = 3 / sin(B) = 12 / sin(C)
By cross-multiplying and solving for sin(A), sin(B), and sin(C), we can then use inverse sine (sin⁻¹) or a calculator to find the values of B and C.
Once you have found the values of all three angles, you will have solved the problem.
say
a = 3
b = 4
c = 1
find one with law of cosines
c^2 = a^2 + b^2 - 2 a b cos C
1 = 9 + 16 - 24 cos C
24 cos C = 24
cos C = 1
C =0
This is not a triangle. It is a straight line because a+c = b