Question

Mr. Gonzales is replacing a cylindrical air-conditioning duct. He estimates the radius of the duct by folding a ruler to form two 6-in. tangents to the duct. The tangents form an angle. Mr. Gonzales measures the angle bisector from the vertex to the duct. It is about 2.75 inches long. What is the radius of the duct?

Answer 1

Answer:

• 5.6875 in

Explanation:

At the point of tangency, the tangent to a circle and the radius form a right triangle (the radius is perpendicular to the tangent).

Here you are given the length of the tangent (6in), and the distance from the bisected vertex to the circle (2.75 in)

I tried to upload the drawing but the tool is not allowing it now.

In the figure:

• The length of the tangent (6 in) is one leg of the triangle
• The distance from vertex and the circle (2.75in)  along with the radius forms the hypotenuse of the right triangle: 2.75 + r.
• The other leg is the radius, r.

Then, you can use Pythagorean theorem:

• (r)² + (6)² = (r +2.75)²

Solve:

• r² + 36 =  r² + 5.5r + 7.5625

• 5.5r = 36 - 7.5625

• 5.5r = 28.4375

• r = 5.6875

The solution is in inches: r = 5.6875 inches ← answer