A triangle has sides A, B, and C. Sides A and B have lengths of 7 and 6, respectively. The angle between A and C is #(11pi)/24# and the angle between B and C is # (11pi)/24#. What is the area of the triangle?

Question

A triangle has sides A, B, and C.  Sides A and B have lengths of 7 and 6, respectively.  The angle between A and C is #(11pi)/24# and the angle between B and C is # (11pi)/24#.  What is the area of the triangle?

Zoe Austin · Accepted Answer

#color(green)(Delta " " A_t = (1/2) a b sin C = 5.44 " sq units"#

#hat A = (11pi)/24, hat B = (11pi)/24, a = 7, b = 6#

#hat C = pi - (11pi)/24 - (11pi)/24 = pi/12#

#color(blue)("Using two sides and included angle formula"#

#color(green)(Delta " " A_t = (1/2) a b sin C = (1/2) * 7 * 6 * sin (pi/12) = 5.44 " sq units"#

Madelyn Adams · Answer

undefined To find the area of the triangle, you can use the formula for the area of a triangle given two sides and the angle between them. The formula is:

$$ \text{Area} = \frac{1}{2} \times \text{side} \times \text{side} \times \sin(\text{angle}) $$

Given that sides A and B have lengths of 7 and 6, respectively, and the angle between A and C as well as B and C is $ \frac{11\pi}{24} $, you can calculate the area using these values. First, find side C using the Law of Cosines:

$$ C^2 = A^2 + B^2 - 2AB \times \cos(\text{angle}) $$

Then, use the formula for the area of the triangle:

$$ \text{Area} = \frac{1}{2} \times A \times B \times \sin(\text{angle}) $$

Substitute the values of A, B, and the angle between them into the formula to find the area of the triangle.