Given that AC and AE are common external tangents of circle G and circle D, FE = 15, GF = 3, and AG = 5, which is the length of AC?

Where is F?

What do you mean by FG and AG? Is G the center of circle G? Specify the points a bit more completely.

Given that AC and AE are common external tangents of circle G and circle D, FE = 15, GF = 3, and AG = 5, which is the length of AC?

26

To find the length of AC, we can use the properties of tangents and apply some geometrical reasoning.

First, let's draw a diagram to visualize the problem.

F G
/ \
/ \
/ \
A ------ E
\ /
\ /
\ /
C

We are given that AC and AE are external tangents of circle G and circle D. This means that AC and AE are perpendicular to GF and FE at points F and E, respectively.

Since AF and AE are perpendicular to GF and FE, we can conclude that the quadrilateral AFEG is a rectangle. This allows us to apply the properties of rectangles to solve the problem.

Since FE = 15 and GF = 3, we can determine the width of the rectangle as follows:
Width of AFEG = GF + FE = 3 + 15 = 18

We are also given that AG = 5, which represents the length of the rectangle.

To find the length of AC, we need to determine the height of the rectangle. Let's call the height HC.

Using the Pythagorean theorem, we can set up the following equation:
AG^2 = AF^2 + FG^2 (applying Pythagoras to triangle AFG)

Substituting the given values:
5^2 = AF^2 + 3^2
25 = AF^2 + 9
16 = AF^2

Taking the square root of both sides, we find that AF = 4.

Since AF and HC are parallel lines, we can conclude that the length of AC is equal to the length of HC. Therefore, AC = HC = 4.

Therefore, the length of AC is 4.