Question

if /_\CDE has sides that are tangent to the circle, and CF = 15 cm, BD = 12 cm, EG = 4 cm. What is the perimeter of CDE?

Answer 1

To determine the perimeter of \( \triangle CDE \) given that it has sides tangent to the inscribed circle, we can utilize the lengths of the segments defined by the points where the tangents meet the sides of the triangle.

Let:
- The tangent segment from point \( C \) to the points of tangency \( D \) and \( E \) be \( CD \) and \( CE \) respectively.
- The tangent segments from \( D \) to the points of tangency \( E \) and \( C \) be \( DE \) and \( DC \).
- The tangent segments from \( E \) to the points of tangency \( C \) and \( D \) be \( EC \) and \( ED \).

Given that \( CF = 15 \, \text{cm} \), \( BD = 12 \, \text{cm} \), and \( EG = 4 \, \text{cm} \), we can represent each tangent segment as follows:
- Let \( CD = x \)
- \( DE = y \)
- \( EC = z \)

Since \( \triangle CDE \) has sides tangent to the circle, by the properties of tangents drawn from an external point, we have:
1. \( CF = CE = x \) (from \( C \))
2. \( BD = BC = y \) (from \( D \))
3. \( EG = AE = z \) (from \( E \))

From the information provided:
- \( CF = 15 \Rightarrow CE = x = 15 \)
- \( BD = 12 \Rightarrow DC = y = 12 \)
- \( EG = 4 \Rightarrow EA = z = 4 \)

Now, to find the perimeter \( P \) of triangle \( CDE \):
\[
P = CD + DE + EC = x + y + z = 15 + 12 + 4 = 31 \, \text{cm}
\]

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