1. The radius of the circumscribed circle of the triangle ABC is 15 cm. Given that B is a 49-degree angle, find the length of side AC.
2. The radius of the circumscribed circle of the triangle ABC is r cm.
Given that B is a â-degree angle, find the length of side AC, in terms of r and â.
To find the length of side AC in both cases, we can use the Law of Sines:
1. In the first case, we have the radius of the circumscribed circle (r) and the angle B (49 degrees). We need to find side AC.
Using the Law of Sines, we have:
sin(A) / AC = sin(B) / BC
Since we know the radius of the circumscribed circle is 15 cm, we can substitute BC with 2r (the diameter of the circle):
sin(A) / AC = sin(B) / (2r)
Next, we can rearrange the equation to solve for AC:
AC = (sin(A) * 2r) / sin(B)
Now, we need to find sin(A). Since we know that the sum of the angles in a triangle is 180 degrees, we can calculate angle A:
A = 180 - (angle B + angle C)
A = 180 - (49 + 90)
A = 41 degrees
Substituting the values into the equation for AC:
AC = (sin(41) * 2 * 15) / sin(49)
Using a calculator, we can evaluate this expression and find the length of side AC.
2. In the second case, we have the radius of the circumscribed circle (r) and the angle B (â degrees). We need to find side AC, in terms of r and â.
Using the Law of Sines, we have:
sin(A) / AC = sin(B) / BC
Since we know the radius of the circumscribed circle is r cm, we can substitute BC with 2r (the diameter of the circle):
sin(A) / AC = sin(B) / (2r)
Next, we can rearrange the equation to solve for AC:
AC = (sin(A) * 2r) / sin(B)
Now, we need to find sin(A). Since we know that the sum of the angles in a triangle is 180 degrees, we can calculate angle A:
A = 180 - (angle B + angle C)
A = 180 - (â + 90)
A = 90 - â degrees
Substituting the values into the equation for AC:
AC = (sin(90 - â) * 2r) / sin(â)
The length of side AC is given in terms of r and â.