a We are asked to find the circumference of a circle given that it is circumscribed about a right triangle whose legs are 12 and 16 inches. Let's sketch a diagram of this situation.
As we can see, to find the diameter of our circle we can use the Pythagorean Theorem. According to this theorem the sum of the squared legs of a right triangle is equal to its squared hypotenuse.
d^2= 12^2+ 16^2
Let's solve the above equation. Since d represents length we will consider only the positive case when taking a square root of d^2.
The diameter of the circle is 20. Finally, let's recall that the circumference of a circle is the product of its diameter and pi.
C= 20π≈ 62.8
The circumference of the circle is 20π, or approximately 62.8 inches.
b In this part we want to evaluate the circumference of a circle that is circumscribed about a square that has a side length of 6 centimeters. Let's draw a diagram describing this situation.
In a square all angles are right, so the diagonal of the square divides it into two right isosceles triangles. Therefore, by the Inscribed Right Triangle Theorem we can say that the hypotenuse of this triangle is also the diameter of the circle, which we will call d.
To find the value of d we will use the Pythagorean Theorem. Since d represents a diameter, we will consider only the positive case when taking a square root of d^2.
The diameter of the circle is approximately 8.5 centimeters. Finally, we will find the circumference of the circle using the fact that the circumference is the product of the diameter and pi.
C= 8.5π≈ 26.7
The circumference of the circle is approximately 26.7 centimeters.
c This time we want to find the circumference of a circle that is inscribed in an equilateral triangle with a side length of 9 inches. Let's look at the picture.
Notice that when a circle is inscribed in a figure each side of the polygon is tangent to this circle.
Next, if we connect each vertex of the triangle with the center of the circle then our triangle will be divided into six congruent 30^(∘)-60^(∘)-90^(∘) triangles.
Now let's focus on the one of the small triangles. Notice that the shorter leg of this triangle is the radius of the circle, which we will call r. Additionally, the longer leg is half of the side length of an equilateral triangle, so it has a length of 92= 4.5.
Since in a 30^(∘)-60^(∘)-90^(∘) the longer leg is sqrt(3) times the shorter leg, we can write an equation for our triangle.
4.5= rsqrt(3) ⇓ r=4.5/sqrt(3)≈ 2.6
The radius of the circle is approximately 2.6 inches. Finally, let's recall that the circumference of a circle is two times the product of its radius and pi.
C=2π( 2.6)≈ 16.3
The circumference of the circle is approximately 16.3 inches.
