For the triangle shown, find the length AD. (Assume u = 19, v = 19, �Úx = 25�‹, and �Úy = 25�‹. Round your answer to two decimal places.)
To find the length AD, we can use the Law of Cosines. The Law of Cosines states that for any triangle with sides a, b, c, and angle C opposite side c, the following formula holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we know the lengths of two sides (u = 19 and v = 19) and the measure of the angle opposite the side we want to find (angle DAB = angle D + angle C = 25 + 25 = 50 degrees).
Using the Law of Cosines:
AD^2 = u^2 + v^2 - 2uv * cos(DAB)
AD^2 = 19^2 + 19^2 - 2(19)(19) * cos(50)
AD^2 = 722 + 722 - 2(19)(19) * cos(50)
AD^2 = 1444 - 722 * cos(50)
Now we can substitute the values into the equation and solve for AD.
To find the length AD of the triangle, we can use the Law of Cosines. The Law of Cosines states that in a triangle with sides a, b, and c, and angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab*cos(C)
In our triangle, we have side AB = u, side BC = v, and the angle C opposite side AC = x. Let's plug in the values into the equation:
AD^2 = AB^2 + BC^2 - 2*AB*BC*cos(C)
AD^2 = u^2 + v^2 - 2*u*v*cos(x)
Now we can substitute the given values:
AD^2 = 19^2 + 19^2 - 2*19*19*cos(25°)
To find the length AD, we need to evaluate this equation:
AD = sqrt(19^2 + 19^2 - 2*19*19*cos(25°))
Using a calculator, we can find the value of AD by plugging in the numbers:
AD ≈ sqrt(19^2 + 19^2 - 2*19*19*cos(25°)) ≈ 19.30
So, the length AD of the triangle is approximately 19.30 units (rounded to two decimal places).