Suppose a triangle has two sides of length 3 and 4 and that the angle
between these two sides is . What is the length of the third side of the
triangle?
A.
B.
C. 3
D.

To find the length of the third side of the triangle, you can apply the Law of Cosines, which states that for any triangle with sides , , and , and the angle opposite side :

In this case, let , , and . Plugging in the values:

Since :

Taking the square root of both sides, we find:

So the length of the third side is , corresponding to option B.

Now, why is the Law of Cosines so great? It combines geometry and algebra, making it applicable for triangles where it’s tough just to rely on sine or cosine functions alone. This means no matter the triangle’s shape, you can find unknown sides or angles with confidence!

Want to impress your friends? Try using the Law of Cosines to create your own origami triangle! Simply print out paper triangles with sides that match your calculations, fold them, and watch geometry come to life in 3D. Then, pull out a protractor and measure the angles! Perfect for a fun afternoon!