Question

A circle with radii marked is inside a triangle.

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

Answer 1

To find the perimeter of triangle \( \triangle CDE \), we need to use the lengths of the segments tangent to the circle.

In triangle geometry involving a circle inscribed within a triangle, the tangents from a point to the circle are equal in length. Let's denote:

- \( s_1 \) as the length of the tangent from point \( C \) to the points where the circle touches sides \( CD \) and \( CE \).
- \( s_2 \) as the length of the tangent from point \( D \) to the points where the circle touches sides \( DE \) and \( DC \).
- \( s_3 \) as the length of the tangent from point \( E \) to the points where the circle touches sides \( EC \) and \( ED \).

The lengths provided are:

- \( CF = 15 \) cm
- \( BD = 12 \) cm
- \( EG = 4 \) cm

From the tangents, we can write:

1. \( CD = CF = 15 \) cm
2. \( DE = BD = 12 \) cm
3. \( EC = EG = 4 \) cm

However, we are not directly concerned with these tangent segments but rather with the equal segments that can be established from the points of tangency.

Let \( x \) be the length of the segment from \( D \) to the point of tangency on \( CE \) (next to point \( E \)), and modify it for others. We can summarize the segments as follows:

- From \( C \), the tangents to the circle touch at \( CD \) and \( CE \): \( s_C = CF = 15 \) cm
- From \( D \), the tangents touch at \( DE \) and \( DC \): \( s_D = BD = 12 \) cm
- From \( E \), the tangents touch at \( EC \) and \( ED \): \( s_E = EG = 4 \) cm

Using this, we find:

\[
CD + DE + EC = s_C + s_D + s_E
\]

Thus,

\[
\text{Perimeter of } \triangle CDE = (s_C + s_E) + (s_D + s_E) + (s_E + s_C)
\]

Since we have the segments already calculated:

\[
\text{Perimeter of } \triangle CDE = CF + BD + EG + CF + BD + EG
\]
This would mean:

\[
15 + 12 + 4 + 15 + 12 + 4 = 2(CF + BD + EG)
\]


Calculating that value:

\[
= 2(15 + 12 + 4) = 2(31) = 62 \text{ cm}
\]

Thus, the perimeter of \( \triangle CDE \) is \( 62 \, \text{cm} \).