find the area of a circle inscribed in triangle ABC where a=9, b=13, and the measure of angle C=38 degrees
You can get c with the law of cosines.
Then, draw the figure. You have a multitude of similar triangles. The idea is to use them to find the measure of radius, the distance on many of those triangles.
I've tried that multiple times, but I was told the answer should be more than one. I get something around .07
As suggested by bobpursley, use the cosine law to get c, I got 8.1
Now use the Sine Law to find angle A, which I got to be 43.16º
Let P be the centre of the incsribed triangle. A property of that circle is that its centre is the intersection of the angle bisectors of the triangle.
Consider triangle APC. Angle C is 19º, AC = 13 and angle A is 21.58º
(If you have a triangle with its base known and the two angles on that base A and C are known, then the height is given by
Height = base/(cot A + cot C)
in this case
height = 13/(cot19º + cot21.58º)
= 2.393
This is the radius of your circle, so the area is pi(2.393)^2
= 17.99 or 18 units
To find the area of a circle inscribed in a triangle, we need to use the concept of the incenter of a triangle. The incenter is the center of the inscribed circle and is equidistant from all three sides of the triangle.
Given the lengths of the sides of triangle ABC (a=9 and b=13) and the measure of angle C (38 degrees), we can start by applying the Law of Cosines to find the length of side c.
The Law of Cosines states that c² = a² + b² - 2ab * cos(C)
Plugging in the given values:
c² = 9² + 13² - 2 * 9 * 13 * cos(38°)
Now, we can calculate c:
c = √(9² + 13² - 2 * 9 * 13 * cos(38°))
Using the formula for the area of a triangle (A = (√(s * (s - a) * (s - b) * (s - c)))), where s is the semiperimeter of the triangle (s = (a + b + c) / 2), we can find the area of triangle ABC.
First, calculate the semiperimeter:
s = (9 + 13 + c) / 2
Then, calculate the area of triangle ABC:
A = √(s * (s - a) * (s - b) * (s - c))
Finally, since the incenter is equidistant from the sides of the triangle, the radius of the inscribed circle can be found using the formula r = A / s.
Once you have the radius, you can calculate the area of the circle using the formula for the area of a circle, A = πr².
Now, let's calculate the values step by step:
Step 1: Calculate the length of side c:
c = √(9² + 13² - 2 * 9 * 13 * cos(38°))
Step 2: Calculate the semiperimeter:
s = (9 + 13 + c) / 2
Step 3: Calculate the area of triangle ABC:
A = √(s * (s - 9) * (s - 13) * (s - c))
Step 4: Calculate the radius of the inscribed circle:
r = A / s
Step 5: Calculate the area of the circle:
Area = πr²
By plugging in the values obtained from these steps, you will find the area of the circle inscribed in triangle ABC.