Question

If ∆ CDE has slides that are tangent to the circle, and CF = 15 cm, BD = 12 cm, and EG = 4 cm, then what is the perimeter of ∆ CDE?

__ cm

Answer 1

To find the perimeter of triangle \( \Delta CDE \), we first note that the lengths \( CF \), \( BD \), and \( EG \) represent the lengths of the segments from the triangle vertices to the points where the internal tangents touch the incircle.

1. **Understanding the tangent segment lengths:**
When a circle is tangent to the sides of a triangle, the length of the tangent segment from each vertex of the triangle to the point where the incircle touches the side is called a tangent segment.
- Let \( A \), \( B \), and \( C \) be the vertices of triangle \( \Delta CDE \) being represented as:
- \( CF \) is the tangent segment from vertex \( C \) to side \( DE \).
- \( BD \) is the tangent segment from vertex \( B \) to side \( CE \).
- \( EG \) is the tangent segment from vertex \( E \) to side \( CD \).

2. **Assigning vertex names to the segments:**
Let's say:
- Let \( A = C \), \( B = D \), and \( C = E \) (or whichever assignment that properly correlates with the given triangle).
- Assume \( C \) is opposite \( DE \), \( D \) is opposite \( CE \), and \( E \) is opposite \( CD \).

3. **Relationship between the tangent segments:**
The lengths \( a, b, c \) representing the sides of the triangle opposite to vertices \( A, B, C \) can be calculated as follows:
\[
a = CF + EG
\]
\[
b = BD + CG
\]
\[
c = AE + CF
\]

However, in our case, let's just utilize the fact that the total perimeter \( P \) of triangle \( \Delta CDE \) is:
\[
P = CF + BD + EG
\]

4. **Substituting the given values:**
- \( CF = 15 \, \text{cm} \)
- \( BD = 12 \, \text{cm} \)
- \( EG = 4 \, \text{cm} \)

5. **Calculating the perimeter:**
\[
P = CF + BD + EG = 15 + 12 + 4
\]
\[
P = 31 \, \text{cm}
\]

Thus, the perimeter of triangle \( \Delta CDE \) is \( \boxed{31} \) cm.