Find the radius of circle 𝑀 given 𝐴𝐶 is equal to 14 centimeters and angle 𝐴 is equal to 50 degrees. Give the answer to two decimal places.
The line 𝐴𝐵 is a diameter of the circle. Therefore, the radius will be half of this length. We also know that the angle 𝐶 is equal to 90 degrees. This is because any angle in a semicircle is equal to 90 degrees.
If we now consider the triangle 𝐴𝐵𝐶, we can use right-angle trigonometry or SOHCAHTOA to calculate the length of 𝐴𝐵. The length 𝐵𝐶 is opposite the 50-degree angle. The length 𝐴𝐶, which we’re told is 14 centimeters, is adjacent to the 50-degree angle. Finally, the length 𝐴𝐵 is the hypotenuse as it is the longest side of the right-angle triangle.
We know the length of the adjacent. And we’re trying to calculate the hypotenuse. Therefore, we will use the cosine ratio. This states that the cos of angle 𝜃 is equal to the adjacent divided by the hypotenuse. Substituting in our values from the diagram gives us cos 50 is equal to 14 divided by the length 𝐴𝐵.
Multiplying both sides of this equation by 𝐴𝐵 and dividing by cos 50 tells us that the length 𝐴𝐵 is equal to 14 divided by cos 50. Typing this into the calculator gives us a value for 𝐴𝐵 equal to 21.7801 and so on. We can therefore say that the diameter of circle 𝑀 is 21.78 centimeters to two decimal places. As the radius is half of the diameter, we can divide this number by two. 21.78 divided by two is equal to 10.89.
This means that the radius of circle 𝑀 is 10.89 centimeters to two decimal places.