Given: ∆ABC, m∠A = 35°
Circle k(O, r=3)
O∈ AB
Find: Perimeter of ∆ABC
arc length s = rθ
so, the perimeter of ABC is 2r+rθ = 2*3 + 3*(35/360)(2π)
Not quite sure what O has to do with it. Maybe there is more that I have not discerned, and you can finish it up.
I have assumed that the central angle C is 35°. Rereading the post, I have some doubts about that. Is ABC just some arbitrary triangle? Is AB an arc of the circle with center at O? But you say O is on the line? arc? AB.
To find the perimeter of ∆ABC, we need to know the lengths of all three sides of the triangle.
Given that O lies on side AB, it indicates that ∆ABC is a right triangle with right angle at O. Since we know the measure of angle A is 35°, we can conclude that angle B is a complementary angle and measures 90° - 35° = 55°.
Now, let's calculate the length of side AB. We're given that O belongs to circle k, which has a radius r = 3. Since O is on AB and O is the center of the circle, we can conclude that the length of AO is equal to the radius, which is 3. By the Pythagorean theorem, we can find the length of side AB:
AB^2 = AO^2 + OB^2
AB^2 = 3^2 + OB^2
AB^2 = 9 + OB^2
Since AB is a side of ∆ABC, we'll call OB simply as 'x'. So the equation becomes:
AB^2 = 9 + x^2
Now, let's find the length of BC. Since BC is the hypotenuse of the right triangle ∆ABC, we can use the Pythagorean theorem once again:
BC^2 = AB^2 + AC^2
BC^2 = AB^2 + AC^2
BC^2 = AB^2 + (AB × tan(A))^2
BC^2 = AB^2 + (AB × tan(35°))^2
BC^2 = AB^2 + (AB × tan(35°))^2
We know AB^2 = 9 + x^2 from the previous equation, so we can substitute it in:
BC^2 = 9 + x^2 + (AB × tan(35°))^2
To find the length of BC, we'll simplify the equation a bit more:
BC^2 = 9 + x^2 + (AB × tan(35°))^2
BC^2 = 9 + x^2 + (AB^2 × tan^2(35°))
BC^2 = 9 + x^2 + (9 + x^2) × tan^2(35°)
BC^2 = 9 + x^2 + (9 + x^2) × tan^2(35°)
Now that we have the lengths of all sides, we can find the perimeter of ∆ABC:
Perimeter = AB + BC + AC
Substituting the values we've calculated:
Perimeter = √(9 + x^2) + √(9 + x^2 + (9 + x^2) × tan^2(35°)) + √x^2
Simplifying this equation further can be done if the value of x is provided.