Recall that in equilateral triangles all angles have measures of 60^(∘), and the isosceles right triangles are 45^(∘)-45^(∘)-90^(∘) triangles. Let x represent each of the missing angles.
As we can see, in the center of the pinwheel we have a full angle, which means that the sum of the measures of all angles is 360^(∘).
3*60^(∘)+3*45^(∘)+3x= 360^(∘)
Let's solve the above equation for x.
The measure of each of the missing angles is 15^(∘).
b In this part we are asked to evaluate the perimeter of the pinwheel. To do this, we will use the fact that the altitude of the blue triangle is 4 inches and that the hypotenuse of the red triangle is congruent to a side of the blue triangle. Let a represents the side length of the blue triangle.
Notice that the perimeter of this pinwheel is three times the perimeter of the wing that is made from one blue and one red triangle. Therefore, from now we can focus on one pair of triangles. Let's label the missing sides.
The perimeter of this figure will be the sum of its outer sides.
P= a+ a+ a+b+d
Notice that the value of d is the difference between the side lengths of blue and red triangles.
d= a-b
We can use this information to simplify the perimeter equation.
Therefore, we only need to find the value of a to evaluate the perimeter of this figure. First let's recall that the altitude in a equilateral triangle divides it into two 30^(∘)-60^(∘)-90^(∘) triangles. This means that 4, which is the longer leg of the 30^(∘)-60^(∘)-90^(∘) triangle, is sqrt(3) times the length of its shorter leg, c.
The value of c is 4sqrt(3)3. Next, let's recall that in 30^(∘)-60^(∘)-90^(∘) triangles the length of the hypotenuse, a, is two times the length of the shorter leg, 4sqrt(3)3.
a=2*4sqrt(3)/3=8sqrt(3)/3
Using this value, we can evaluate the perimeter of one pair of triangles.
