In triangle ABC, cos A = 2/3 and cos B = 2/9. Given the perimeter of the triangle ABC is 24, compute the area of triangle ABC.

Guest Aug 3, 2022

#1**+2 **

Find angle A = cos^{-1}(2/3) = 48.19 (Deg), and angle B = cos^{-1}(2/9) = 77.16 (Deg), so angle C = 180 - 48.19 - 77.16 = 54.65 (Deg).

The Sine Rule states that a/sin(A) = b/sin(B) = c/sin(C). We also know that a + b + c = 24.

Let's try and use both of these to express a and b by means of c.**a** = c x (sin(A)) / sin(C) = c * 0.745 / 0.816 = 0.913 x c,** b** = c x (sin(B)) / sin(C) = c x 0.975 / 0.816 = 1.195 x c.

Substituting (exchanging) a and b for these gives us that: a + b + c = 24 is the same as 0.913 c + 1.195 c + c = 24,

or 3.108 c = 24. Divding on both sides gives us c = 24 / 3.108 = 7.722.

Now we can back-substitute and find a and b,** a** = 0.913c = 0.913 x 7.722 = 7.050, and **b** = 1.195c = 1.195 x 7.722 = 9.228.

We can find the area of a trangle given all it's sides by using Heron's formula:

(Se: https://www.mathsisfun.com/geometry/herons-formula.html )

We find s = 1/2 x ( a + b + c ) = 1/2 (7.050 + 9.228 + 7.722) = 1/2 (24) = 12.

[This being a whole number indicates we are on the right path)

The area is given by \(A =\sqrt{s(s-a)(s-b)(s-c)}\) , so \(A =\sqrt{12\times4.59\times2.772\times4.278} = \sqrt{704.402} \approx 26.54\)

Hope this helps!

tuffla2022 Aug 3, 2022

#1**+2 **

Best Answer

Find angle A = cos^{-1}(2/3) = 48.19 (Deg), and angle B = cos^{-1}(2/9) = 77.16 (Deg), so angle C = 180 - 48.19 - 77.16 = 54.65 (Deg).

The Sine Rule states that a/sin(A) = b/sin(B) = c/sin(C). We also know that a + b + c = 24.

Let's try and use both of these to express a and b by means of c.**a** = c x (sin(A)) / sin(C) = c * 0.745 / 0.816 = 0.913 x c,** b** = c x (sin(B)) / sin(C) = c x 0.975 / 0.816 = 1.195 x c.

Substituting (exchanging) a and b for these gives us that: a + b + c = 24 is the same as 0.913 c + 1.195 c + c = 24,

or 3.108 c = 24. Divding on both sides gives us c = 24 / 3.108 = 7.722.

Now we can back-substitute and find a and b,** a** = 0.913c = 0.913 x 7.722 = 7.050, and **b** = 1.195c = 1.195 x 7.722 = 9.228.

We can find the area of a trangle given all it's sides by using Heron's formula:

(Se: https://www.mathsisfun.com/geometry/herons-formula.html )

We find s = 1/2 x ( a + b + c ) = 1/2 (7.050 + 9.228 + 7.722) = 1/2 (24) = 12.

[This being a whole number indicates we are on the right path)

The area is given by \(A =\sqrt{s(s-a)(s-b)(s-c)}\) , so \(A =\sqrt{12\times4.59\times2.772\times4.278} = \sqrt{704.402} \approx 26.54\)

Hope this helps!

tuffla2022 Aug 3, 2022