Problem 10.012: From point A, not lying on the circle, a tangent and a secant are drawn to it (Fig. 10.12). The distance from point A to the point of tangency is 16 cm, and to one of the intersection points of the secant with the circle is 32 cm. Find the radius of the circle if the secant is 5 cm away from its center.
Solution Explanation:
Hello! Today we’re going to solve an interesting geometry problem involving a circle, a tangent, and a secant. We’ll break down each step to make it clear.
Step 1: Understand what is given and what needs to be found.
First, let’s identify what information we have and what our goal is.
- Given:
- Tangent segment AB = 16 cm.
- Secant segment AC = 32 cm.
- Distance from the circle’s center (O) to the secant (OE) = 5 cm.
- To Find: The radius of the circle (e.g., segment OC).
Step 2: Calculate the length of segment AD.
What we are calculating: The length of segment AD, which is the external part of the secant.
Purpose of this step: We use the Tangent-Secant Theorem, which states that the square of the tangent segment’s length equals the product of the entire secant segment’s length and its external part. This relation is AB^2 = AC \cdot AD. Finding AD will allow us to determine the length of the chord CD.
Calculation:
16^2 = 32 \cdot AD 256 = 32 \cdot ADAD = \frac{256}{32} = 8 cm.
Step 3: Calculate the length of segment CE.
What we are calculating: The length of segment CE, which is half of the chord CD.
Purpose of this step: A perpendicular drawn from the center of a circle to a chord bisects the chord. So, E is the midpoint of CD. Knowing CE and the given OE (5 cm), we can form a right-angled triangle OEC and use the Pythagorean theorem to find the radius OC.
Calculation:
First, find the length of chord CD: CD = AC - AD = 32 - 8 = 24 cm.
Then, CE = \frac{CD}{2} = \frac{24}{2} = 12 cm.
Step 4: Calculate the radius of the circle, CO.
What we are calculating: The radius of the circle, segment CO.
Purpose of this step: This is the final step to find the answer to our problem.
Calculation:
Consider the right-angled triangle OEC (since OE is perpendicular to the secant). The legs are OE = 5 cm and CE = 12 cm. The hypotenuse CO is the radius.
Applying the Pythagorean theorem: CO^2 = OE^2 + CE^2
CO^2 = 5^2 + 12^2 CO^2 = 25 + 144 CO^2 = 169CO = \sqrt{169} = 13 cm.
