𝐴𝐵𝐶 is a triangle where 𝑃 minus 𝑎 is equal to 20 centimeters, 𝑃 plus 𝑎 is equal to 116 centimeters, 𝑏 is equal to 41 centimeters, and the perimeter is two 𝑃. Find the measure of the smallest angle in 𝐴𝐵𝐶, giving the answer to the nearest second.
Sketching a triangle 𝐴𝐵𝐶, we are told that side length 𝑏 is 41 centimeters. We are also given two equations: 𝑃 minus 𝑎 is equal to 20 centimeters and 𝑃 plus 𝑎 is equal to 116 centimeters. We can solve these simultaneously by firstly adding equation one and equation two. This gives us two 𝑃 is equal to 136. Dividing both sides of this equation by two gives us 𝑃 is equal to 68. As we are told the perimeter of the triangle is two 𝑃, this is equal to 136 centimeters. Substituting 𝑃 equals 68 into equation two gives us 68 plus 𝑎 is equal to 116. We can subtract 68 from both sides of this equation such that 𝑎 is equal to 48. Side length 𝑎, which is opposite the angle capital 𝐴, is equal to 48 centimeters.
We can then calculate side length 𝑐 by subtracting 48 and 41 from 136. This is equal to 47. The side opposite angle 𝐶 is 47 centimeters. We are asked to calculate the measure of the smallest angle. And we know in any triangle the smallest angle is opposite the smallest side. This means that angle 𝐵, which we have labeled 𝜃, will be the smallest angle in the triangle. We can calculate this using the law of cosines or cosine rule, which states that the cos of angle 𝐴 is equal to 𝑏 squared plus 𝑐 squared minus 𝑎 squared all divided by two multiplied by 𝑏 multiplied by 𝑐. It is important to note that the side length that we are subtracting is the one opposite our angle.
In this question, we have cos 𝜃 is equal to 48 squared plus 47 squared minus 41 squared all divided by two multiplied by 48 multiplied by 47. The right-hand side of our calculation simplifies to 59 over 94. We can then take the inverse cos of both sides of our equation so that 𝜃 is equal to the inverse cos of 59 over 94. Typing this into the calculator, we get 𝜃 is equal to 51.1223 and so on degrees. As we are asked to give our answer to the nearest second, this is equal to 51 degrees, seven minutes, and 20 seconds. This is the smallest of the three angles in the triangle 𝐴𝐵𝐶.