Questions: En la figura, AB=6 cm, OC=9 cm y OB=3 cm. ¿cuánto mide CD?

Transcript text: En la figura, $A B=6 \mathrm{~cm}, O C=9 \mathrm{~cm}$ y $O B=3 \mathrm{~cm}$. ¿cuánto mide $C D$ ?

Solution

Solution Steps

Step 1: Identify Given Information

We are given the following measurements:

$ AB = 6 $ cm
$ OC = 9 $ cm
$ OB = 3 $ cm

Step 2: Understand the Relationship

The figure shows two intersecting lines $ L_1 $ and $ L_2 $ with point $ O $ as the intersection. We need to find the length of $ CD $.

Step 3: Apply the Intersecting Chords Theorem

The Intersecting Chords Theorem states that if two chords intersect each other inside a circle, the products of the lengths of the segments of each chord are equal. Therefore, we have: \[ AO \cdot OB = CO \cdot OD \]

Step 4: Calculate Segment Lengths

From the given information:

$ AB = 6 $ cm, so $ AO + OB = 6 $ cm
$ OB = 3 $ cm, so $ AO = 6 - 3 = 3 $ cm

Step 5: Set Up the Equation

Using the Intersecting Chords Theorem: \[ AO \cdot OB = CO \cdot OD \] \[ 3 \cdot 3 = 9 \cdot OD \]

Step 6: Solve for $ OD $

\[ 9 = 9 \cdot OD \] \[ OD = 1 \) cm

Step 7: Calculate $ CD $

Since $ CD = CO + OD $: \[ CD = 9 + 1 = 10 \) cm

Final Answer

The length of $ CD $ is $ 10 $ cm.

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