DE is a tangent to the circumcircle of `DeltaABC ` at the vertex A such that `DE||BC`. If `AB = 17 cm`, then the length of AC is equal to

Options

A. `16.0 cm `
B. `16.8 cm`
C. `17.3 cm`
D. `17 cm `

Answer

To solve the problem, we need to use the properties of parallel lines and angles in triangles. Here’s a step-by-step solution:

### Step 1: Understand the configuration
We have a triangle ABC with a circumcircle. DE is a tangent to the circumcircle at point A, and DE is parallel to BC. We know that AB = 17 cm.

### Step 2: Use the properties of parallel lines
Since DE is parallel to BC, we can use the Alternate Segment Theorem. This theorem states that the angle formed between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

### Step 3: Set up the angles
Let’s denote:
- Angle DAB (the angle between the tangent DE and line AB) as θ.
- Angle ACB (the angle opposite to side AB in triangle ABC) as θ as well, since they are equal due to the Alternate Segment Theorem.

### Step 4: Apply the property of triangles
From the properties of triangles, we know that if two angles are equal, the sides opposite to those angles are also equal. Therefore, we can conclude that:
- If angle DAB = angle ACB, then AB = AC.

### Step 5: Substitute the known value
Since we know that AB = 17 cm, we can now state that:
- AC = AB = 17 cm.

### Conclusion
Thus, the length of AC is equal to 17 cm.

### Final Answer
AC = 17 cm.

---