Question

circles, triangles, and points of tangency :
6.) consider the diagram drawn below to find the diameter of the larger circle.

• the diameter of the smaller circle is 12.
• the length of the chord pictured is 16.
• the chord pictured is tangent to the smaller picture.

7.) in the image below ab and cb are tangent to circle o below. if ab measures 8 cm and oa is 6 cm, what is the length of ob?

Explanation:

Problem 6:

Step 1: Find the radius of the smaller circle

The diameter of the smaller circle is 12, so the radius \( r \) (let's call it \( c \)) is \( \frac{12}{2} = 6 \).

Step 2: Find half of the chord length

The chord length is 16, so half of it is \( \frac{16}{2} = 8 \).

Step 3: Use the Pythagorean theorem to find the radius of the larger circle

Let \( R \) be the radius of the larger circle. We have a right triangle with one leg as the radius of the smaller circle (6), another leg as half the chord (8), and the hypotenuse as the radius of the larger circle (\( R \)). By the Pythagorean theorem, \( R^2 = 6^2 + 8^2 \).
Calculating: \( R^2 = 36 + 64 = 100 \), so \( R = \sqrt{100} = 10 \).

Step 4: Find the diameter of the larger circle

The diameter is \( 2R = 2\times10 = 20 \).

Step 1: Recall the property of tangents to a circle

A tangent to a circle is perpendicular to the radius at the point of tangency. So, \( OA \perp AB \), meaning triangle \( OAB \) is a right triangle with \( \angle OAB = 90^\circ \).

Step 2: Apply the Pythagorean theorem

In right triangle \( OAB \), \( OA = 6 \) cm (radius), \( AB = 8 \) cm (tangent segment), and \( OB \) is the hypotenuse. By the Pythagorean theorem, \( OB^2 = OA^2 + AB^2 \).
Calculating: \( OB^2 = 6^2 + 8^2 = 36 + 64 = 100 \), so \( OB = \sqrt{100} = 10 \) cm.

Answer:

20

Problem 7: