To find the area of triangle ABE, we need to find the length of segment AB and the length of segment BE.
First, let's find the length of segment AB. Since circle A is tangent to circle C at point B, we know that AB is equal to the sum of the radii of circle A and circle C. So, AB = radius of A + radius of C = 11 + 8 = 19 meters.
Next, let's find the length of segment BE. Since BE is tangent to circle A, it is perpendicular to radius AB at point B. This makes triangle ABE a right triangle. Using the Pythagorean theorem, we can find the length of BE.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (in this case, AB) is equal to the sum of the squares of the lengths of the other two sides (in this case, AE and BE).
AB^2 = AE^2 + BE^2
Substituting the values we know, we have:
19^2 = 11^2 + BE^2
361 = 121 + BE^2
BE^2 = 361 - 121
BE^2 = 240
Taking the square root of both sides, we get:
BE = √240 ≈ 15.49 meters
Now that we have the lengths of AB and BE, we can find the area of triangle ABE. The area of a triangle can be calculated using the formula A = (1/2) * base * height.
In this case, AB is the base and BE is the height. So, the area of triangle ABE is:
A = (1/2) * AB * BE
= (1/2) * 19 * 15.49
≈ 145.59 square meters
Therefore, the area of triangle ABE is approximately 146 square meters.