We are given that Colleen and Mike are planning a party, and Colleen wants to sew decorations in the shape of isosceles triangles with angle measures of 64^(∘) at the base and side lengths of 5 inches. Let's sketch a diagram describing this situation.
To evaluate the perimeter of one triangle, we need to know all the side lengths. Let x represent the length of the base of this triangle. First, we can evaluate the measure of the third angle, let's call it y, by the Triangle Angle Sum Theorem.
Next, let's recall the Law of Sines. If △ ABC has lengths of a, b, and c and angle measures of A, B, and C, then we can write that the ratios of the sine of an angle to the side opposite this angle are equal.
Using this law, we can create a proportion for x.
sin 64^(∘)/5=sin 52^(∘)/x
We can solve this equation using cross multiplication.
The length of the base of this triangle is approximately 4.4.
As we know all the side lengths, we can evaluate the perimeter of the triangle.
P= 5+ 5+4.4=14.4
The perimeter of one triangle is approximately 14.4. This means that neither Colleen nor Mike is correct.
