Peter has three sticks measuring 19 inches, 23 inches, and 27 inches. He lays form a triangle. Find the measure of the angle enclosed by the 19 inch and 23 insides to the nearest degree.
To find the measure of the angle enclosed by the 19-inch and 23-inch sides of the triangle, we can use the Law of Cosines. The Law of Cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides, minus twice the product of the lengths of those sides and the cosine of the angle between them.
In this case, let's use the 19-inch side as side a, the 23-inch side as side b, and the angle enclosed by them as angle C. The formula for the Law of Cosines is:
c^2 = a^2 + b^2 - 2ab * cos(C)
Now we can substitute the values given in the problem:
c^2 = 19^2 + 23^2 - 2 * 19 * 23 * cos(C)
Simplifying and solving for cos(C), we have:
cos(C) = (19^2 + 23^2 - 27^2) / (2 * 19 * 23)
cos(C) = (361 + 529 - 729) / (2 * 19 * 23)
cos(C) = 161 / (2 * 19 * 23)
cos(C) ≈ 0.374
To find the measure of angle C, we can take the arccosine of 0.374:
C ≈ arccos(0.374)
Using a calculator or an online tool, we find:
C ≈ 68.7 degrees (rounded to the nearest degree)
Therefore, the measure of the angle enclosed by the 19-inch and 23-inch sides of the triangle is approximately 69 degrees.