Two circles with centers B and D have radii DA = 8 cm and BC = x cm, respectively. AC is tangent to both circles. If DB and AC intersect the point E, AE = 12 cm and EC = 18 cm, then find the value of x (in cm).

The correct answer is 12.

Step-by-step Explanation:

Let us analyze the given geometric configuration involving the two circles and their common tangent line:
1. We have two circles centered at D and B with radii DA = 8 cm and BC = x cm, respectively.
2. Line segment AC is tangent to the first circle at point A and to the second circle at point C.
3. By the standard property of circle tangents, a radius drawn to a point of tangency is perpendicular to the tangent line at that point. Therefore:

$\angle DAE=90°$

and

$\angle BCE=90°$

4. Line segment DB connects the centers of the two circles and intersects the tangent line AC at point E.
5. At the intersection point E, the vertically opposite angles formed by lines DB and AC are equal:

$\angle AED=\angle CEB$

Similarity of Triangles:
Now, compare right-angled triangles $△DAE$ and $△BCE$:
- Both triangles have a right angle ($\angle DAE=\angle BCE=90°$).
- They share equal vertically opposite angles ($\angle AED=\angle CEB$).

By the Angle-Angle (AA) similarity criterion:

$△DAE\sim △BCE$

Calculating the value of x:
Since corresponding sides of similar triangles are proportional, we can write the ratio:

$\frac{DA}{BC}=\frac{AE}{EC}$

Substitute the given values into the proportion:
- Radius DA = 8 cm
- Radius BC = x cm
- Length AE = 12 cm
- Length EC = 18 cm

$\frac{8}{x}=\frac{12}{18}$

Simplify the ratio on the right side:

$\frac{12}{18}=\frac{2}{3}$

So, the equation becomes:

$\frac{8}{\mathrm{x<!--\; population--><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac>}}$

Cross-multiplying to solve for x:

$2·x=8·3$

$2x=24$

$x=\frac{24}{2}=12$

Thus, the value of x is 12 cm.