Line AC and AE are common external tangents of point G and D. Line FE=13, line GF=5 and line AG=13.
what is the measure of AC?
To find the measure of line AC, we can use the concept of tangent lines and the properties of tangents to circles.
First, let's draw a diagram to better understand the given information:
C G D F
/ | | |
/ | | |
/____________|_________|________|
A E
Based on the given information, we have line AC and AE as common external tangents of point G and D. This means that lines AC and AE are tangent to the circles centered at G and D.
Now, let's analyze the triangle AFG. We know that line GF is a tangent to the circle centered at G. Also, we're given that line AG has a length of 13 units. Using these pieces of information, we can apply the tangent-chord theorem.
The tangent-chord theorem states that if a line is tangent to a circle, then the length of the tangent line squared is equal to the product of the lengths of the two segments it divides.
In our case, using the tangent-chord theorem with line GF as the tangent, we have:
GF^2 = AG * GD
Substituting the given values:
5^2 = 13 * GD
25 = 13 * GD
Solving for GD, we find:
GD = 25 / 13
Now, let's analyze the triangle ADE. We know that line AE is a tangent to the circle centered at D. Also, we're given that line DE has a length of 13 units. Using these pieces of information, we can again apply the tangent-chord theorem.
Using the tangent-chord theorem with line AE as the tangent, we have:
AE^2 = AG * GD
Substituting the given values:
AE^2 = 13 * GD
AE^2 = 13 * (25 / 13)
AE^2 = 25
Therefore, AE = 5.
Since lines AC and AE are tangents from the same point A to the same circle, they must be equal in length. Therefore, AC = AE = 5.
So, the measure of line AC is 5 units.