In the triangle ABC, the sides AB and BC have lengths 7cm and 8cm respectively, and angle B is obtuse. the area of the triangle is 14 root 3 cm^2. Find
a.The size of angle B
b. The length of AC
formula for area of triangle
= (1/2)ab sinØ, where Ø is the contained angle of sides a and b
(1/2)(7)(8) sinØ = 14√3
sin Ø = 28√3/56 = √3/2
Ø = 60°
now use the cosine law to find AC
given triangle ABC with
BSC IN EE
To find the size of angle B and the length of AC, we can use the Law of Cosines and the formula for the area of a triangle.
a. Finding the size of angle B:
In a triangle, the Law of Cosines states that for any side, c, the following formula holds:
c^2 = a^2 + b^2 - 2ab*cos(C)
In our triangle ABC, we know that AB = 7 cm, BC = 8 cm, and AC is unknown. We also know that the angle B is obtuse.
Let's assume AC = x cm.
Using the Law of Cosines, we have:
x^2 = 7^2 + 8^2 - 2(7)(8)*cos(B)
x^2 = 49 + 64 - 112*cos(B)
x^2 = 113 - 112*cos(B)
To find the value of cos(B), we need to use the fact that the area of the triangle is 14√3 cm^2.
The area of a triangle can be calculated using the following formula:
Area = (1/2) * a * b * sin(C)
In our case, the area is given as 14√3 cm^2. Substituting the given values:
14√3 = (1/2) * 7 * 8 * sin(B)
14√3 = 28 * sin(B)
sin(B) = (14√3) / 28
sin(B) = √3 / 2
Since the angle B is obtuse, sin(B) will be positive. Therefore, we can conclude that B is a 60 degree angle.
Now, substituting cos(B) into the equation for x^2:
x^2 = 113 - 112*(√3 / 2)
x^2 = 113 - 112√3
b. Finding the length of AC:
To find the length of AC, we need to take the square root of x^2:
AC = √(113 - 112√3) cm
Therefore,
a. The size of angle B is 60 degrees.
b. The length of AC is √(113 - 112√3) cm.