A triangle has sides A, B, and C. Sides A and B have lengths of 7 and 5, respectively. The angle between A and C is $\frac{11 π}{24}$ $(11pi)/24$ and the angle between B and C is $\frac{7 π}{24}$ $(7pi)/24$ . 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 5, respectively. The angle between A and C is $\frac{11 π}{24}$ $(11pi)/24$ and the angle between B and C is $\frac{7 π}{24}$ $(7pi)/24$ . What is the area of the triangle?

1 Answer

Impact of this question

1663 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-02-21T12:31:46

${Area}_{△} = \approx 12.97$ $"Area"_triangle=~~12.97$

Explanation:

Call the angle opposite side A as $∠ a$ $/_a$ (i.e. the angle between B and C);
and similarly the angle opposite Side B as $∠ b$ $/_b$
and the angle opposite Side C as $∠ c$ $/_c$

We are told (among other things) that:
$A = 7, ∠ a = \frac{7 π}{24}, and ∠ c = \frac{11 π}{24}$ $A=7, /_a=(7pi)/24, and /_c=(11pi)/24$

The Sine Law tells us
$XXX \frac{C}{sin (c)} = \frac{A}{sin (a)}$ $color(white)("XXX")C/sin(c) = A/sin(a)$

So
$XXX C = \frac{7}{sin (\frac{7 π}{24})} \cdot sin (\frac{11 π}{24})$ $color(white)("XXX")C=7/(sin((7pi)/24))*sin((11pi)/24)$

Evaluating we get: $C = 11$ $C=11$

We now have the lengths of the three sides and can apply Heron's Formula:
$XXX {Area}_{△} = \sqrt{S (A - A) (S - B) (S - C)}$ $color(white)("XXX")"Area"_triangle = sqrt(S(A-A)(S-B)(S-C))$
where $S$ $S$ is the semi-perimeter (i.e. $\frac{A + B + C}{2}$ $(A+B+C)/2$ )

enter image source here

Science

Math

Humanities

... and beyond

A triangle has sides A, B, and C. Sides A and B have lengths of 7 and 5, respectively. The angle between A and C is $\frac{11 π}{24}$ $(11pi)/24$ and the angle between B and C is $\frac{7 π}{24}$ $(7pi)/24$ . What is the area of the triangle?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

A triangle has sides A, B, and C. Sides A and B have lengths of 7 and 5, respectively. The angle between A and C is 11π24(11pi)/24 and the angle between B and C is 7π24 (7pi)/24. What is the area of the triangle?

1 Answer

Explanation:

Related questions

Impact of this question

A triangle has sides A, B, and C. Sides A and B have lengths of 7 and 5, respectively. The angle between A and C is $\frac{11 π}{24}$ $(11pi)/24$ and the angle between B and C is $\frac{7 π}{24}$ $(7pi)/24$ . What is the area of the triangle?