AC and AE are common external tangents of circle G and circle D. FE = 26, GF = 5, and AG =13. What is the measure of AC?
To find the measure of AC, we need to first understand the properties of tangents.
In this problem, AC is a common external tangent of circle G and circle D. This means that AC is tangent to both circles at points A and C.
Now, let's see how we can use the given information to find the measure of AC.
First, we know that FE = 26, GF = 5, and AG = 13. Let's label the point of tangency between circle G and AC as B.
Now, we can apply the tangent-tangent theorem. According to this theorem, when two tangents intersect at an external point, the square of the length of the segment joining the two points of contact is equal to the product of the lengths of the external segments on the respective tangents.
Using this theorem, we can write the following equation:
AB^2 = EB * FB = (FE + EB) * (GF + FB)
We know that FE = 26, GF = 5, and FB = 13 (as AG = 13), so the equation becomes:
AB^2 = (26 + EB) * (5 + 13)
Now, let's solve for AB.
AB^2 = (26 + EB) * 18
AB^2 = 468 + 18EB
Next, we need to consider another tangent segment. Let's label the point of tangency between circle D and AC as D.
The same tangent-tangent theorem applies here. So, we can write the following equation:
AC^2 = AD * CD
We are given that AG = 13, and we can write AG = AD + DG, using triangle AGD.
Substituting AG = 13 and DG = 26 into the equation, we get:
13 * CD = AC^2
Now, let's substitute AB^2 = 468 + 18EB into the equation:
13 * CD = (468 + 18EB)
Finally, we have two equations:
AB^2 = 468 + 18EB
13 * CD = (468 + 18EB)
By solving these two equations simultaneously, we can find the values of AB and CD, and thus determine the measure of AC.
Unfortunately, without additional information about the lengths of EB and CD, we cannot solve for AB or AC.